1 Introduction

We study the optimal control problem

$$\begin{aligned} \text{(P) } \quad \inf _{u \in {U_{ad}}\cap L^\infty (Q)} J(u):= \frac{1}{2}\int _Q (y_u(x,t) - y_d(x,t))^2\,\mathrm {d}x\,\mathrm {d}t+ \frac{\kappa }{2}\int _Q u(x,t)^2\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$

where \(\kappa > 0\),

$$\begin{aligned} {U_{ad}}= \{u \in L^\infty (0,T;L^1(\varOmega )) : \Vert u(t)\Vert _{L^1(\varOmega )} \le \gamma \text { for a.a. } t \in (0,T)\} \end{aligned}$$

with \(0< \gamma < +\infty \), and \(y_u\) is the solution of the semilinear parabolic equation

$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle \frac{\partial y}{\partial t} + Ay + a(x,t,y) = u &{} \text{ in } Q = \varOmega \times (0,T),\\ y = 0 \text{ on } \varSigma = \Gamma \times (0,T),&{} y(0) = y_0 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$
(1.1)

with

$$\begin{aligned} Ay=-\sum _{i,j=1}^{n}\partial _{x_j}(a_{ij}(x)\partial _{x_i}y). \end{aligned}$$

We assume that \(\varOmega \) is a bounded, connected, and open subset of \({\mathbb {R}}^n\), \(n = 2\) or 3, with a Lipschitz boundary \(\Gamma \), and that \(0< T < \infty \) is fixed.

The precise conditions on the nonlinearity a will be given below. Suffice it to say at this moment that strong nonlinearities such as \(\exp (y)\), \(\sin (y)\), or polynomial nonlinearities with positive leading term of odd degree will be admitted. A first difficulty that arises in treating \( \text{(P) } \) relates to the proof of existence of an optimal control. The reader could think of choosing \(L^2(Q)\) as the convenient space to prove the existence of a solution because of the coercivity of J on this space and since the constraint defines a closed and convex subset of \(L^2(Q)\). However, the selection of controls in \(L^2(Q)\) is not appropriate to deal with the non-linearity in the sate equation. Indeed, even if we can prove the existence of a solution of the state equation, its regularity is not enough (it is not an element of \(L^\infty (Q)\), in general) to get the differentiability of the relation control to state. Looking at the control constraint and the cost functional, a second possibility is to consider \(L^\infty (0,T;L^2(\varOmega ))\) as control space. But this is not a reflexive Banach space and, consequently, the proof of existence of a solution to (P) cannot be done by standard techniques. Nevertheless, we can prove existence of solutions in the spaces \(L^r(0,T;L^2(\varOmega ))\) for all \(r > \frac{4}{4-n}\). Moreover, all these solutions belong to \(L^\infty (Q)\). This leads us to formulate the control problem in \(L^\infty (Q)\); see Remark (4.2). To deal with the non-linearity of the state equation in the proof of a solution to (P) in \(L^\infty (Q)\), one approach consists in introducing artificial bound constraints on the control and prove that they are inactive as the artificial constraint parameter is large enough; see, for instance [7]. In our case, this would lead to two control constraints with two Lagrange multipliers in the dual of \(L^\infty \). This makes the proof of boundedness of the optimal control very difficult. In this work we avoid such a technique and rather modify (truncate) the non-linear term of the state equation and prove that for a large truncation parameter the cut off is not active on the optimal state.

A second difficulty results from the non-differentiability of the constraint on the control in the definition of \(U_{ad}\). This is a natural constraint since it models a volumetric restriction, which represents a limit to the total amount of control acting at any time t. This technological constraint is an alternative to pointwise or to energy constraints which have been considered previously in the literature. Moreover, the \(L^1\)-norm in space leads to a spatially sparsifying effect for the solutions. It is different from the type of sparsification which results when considering such terms in the cost. While for the former, sparsification takes place only after the control becomes active, for the latter it takes place regardless of the norm of the control. For problem \( \text{(P) } \) the sparsity effect is described by the level set characterized by the functional values of the adjoint state at the height of the supremum norm of the multiplier associated to the control constraint in (P); see Corollary 3.3. We point out that while the \(L^2\) norm appearing in the cost influences the optimal solution, it does not eliminate the sparsifying effect of \(L^1\)-terms, regardless of whether they appear in the cost or as a constraint. The literature on problems with an \(L^1\) or measure-valued norm in the cost is quite rich, so we can only give selected references which consider evolutionary problems [1,2,3,4,5, 7,8,9, 11, 14,15,16, 18]. In all these papers, either there are no control constraints or they are box constraints. In [13], the authors study a control problem for the evolutionary Navier–Stokes system under the smooth control constraint \(\Vert u(t)\Vert ^2_{L^2(\varOmega )} \le 1\), which is smooth and not sparsifying. In [6], the control of the 2d evolutionary Navier–Stokes system is analyzed, where the controls are measured valued functions subject to the constraint \(\Vert u(t)\Vert _{M(\varOmega )} \le \gamma \).

The structure of the paper is the following. The analysis of the state equation and its first and second derivatives with respect to the controls is carried out in Sect. 2. Here special attention is paid to the \(L^\infty (Q)\) regularity of the state variable. In Sect. 3 first order optimality conditions are derived and the structural properties of the involved functions are analyzed. In particular, the regularity of the optimal control is proved, which is a crucial point for the numerical analysis of the control problem. The proof of existence of an optimal control is given in Sect. 4. Section 5 is devoted to necessary and sufficient second order optimality conditions. In the final section, as a consequence of the second order condition, Hölder and Lipschitz stability of local solutions with respect to the control bound \(\gamma \) is investigated.

2 Analysis of the State Equation

In this section we establish the well posedness of the state equation, the regularity of the solution, and the differentiable dependence of the solution with respect to the control. To this end we make the following assumptions.

We assume that \(y_0 \in L^\infty (\varOmega )\), \(a_{ij} \in L^\infty (\varOmega )\) for every \(1 \le i, j \le n\), and

$$\begin{aligned} \varLambda _A|\xi |^2\le \sum _{i,j=1}^n a_{ij}(x)\xi _i\xi _j\ \ \forall \xi \in {\mathbb {R}}^n \ \text{ for } \text{ a.a. } x\in \varOmega \end{aligned}$$
(2.1)

for some \(\varLambda _A > 0\). We also assume that \(a:Q\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function of class \(C^2\) with respect to the last variable satisfying the following properties:

$$\begin{aligned}&\exists C_a \in {\mathbb {R}} : \frac{\partial a}{\partial y}(x,t,y)\ge C_a\ \forall y\in {\mathbb {R}}, \end{aligned}$$
(2.2)
$$\begin{aligned}&a(\cdot ,\cdot ,0)\in L^{{{\hat{r}}}}(0,T;L^{{{\hat{p}}}}(\varOmega )), \text { with } {{\hat{r}}}, {{\hat{p}}}\ge 2 \ \text { and }\ \frac{1}{{{\hat{r}}}} + \frac{n}{2{{\hat{p}}}} < 1, \end{aligned}$$
(2.3)
$$\begin{aligned}&\forall M>0\ \exists C_{a,M}>0 : \left| \frac{\partial ^j a}{\partial y^j}(x,t,y)\right| \le C_{a,M}\ \forall |y|\le M \text{ and } j=1,2, \end{aligned}$$
(2.4)
$$\begin{aligned}&\begin{array}{l}\!\!\!\forall \rho>0 \text { and } \forall M>0\ \exists \varepsilon >0 \text { such that}\\ \displaystyle \!\!\!\left| \frac{\partial ^2 a}{\partial y^2}(x,t,y_1)-\frac{\partial ^2 a}{\partial y^2}(x,t,y_2)\right|<\rho \ \ \ \forall |y_1|,|y_2|\le M \text { with } \ |y_1-y_2|<\varepsilon ,\end{array} \end{aligned}$$
(2.5)

for almost all \((x,t) \in Q\).

As usual W(0, T) denotes the Hilbert space

$$\begin{aligned} W(0,T) = \{y \in L^2(0,T;H_0^1(\varOmega )) : \frac{\partial y}{\partial t} \in L^2(0,T;H^{-1}(\varOmega ))\}. \end{aligned}$$

We recall that W(0, T) is continuously embedded in \(C([0,T];L^2(\varOmega ))\) and compactly embedded in \(L^2(Q)\).

Theorem 2.1

Under the previous assumptions, for every \(u \in L^r(0,T;L^p(\varOmega ))\) with \(\frac{1}{r} + \frac{n}{2p} < 1\) and \(r, p \ge 2\) there exists a unique solution \(y_u \in L^\infty (Q) \cap W(0,T)\) of (1.1). Moreover, the following estimates hold

$$\begin{aligned}&\Vert y_u\Vert _{L^\infty (Q)} \le \eta \big (\Vert u\Vert _{L^r(0,T;L^p(\varOmega ))} + \Vert a(\cdot ,\cdot ,0)\Vert _{L^{{\hat{r}}}(0,T;L^{{\hat{p}}}(\varOmega ))} + \Vert y_0\Vert _{L^\infty (\varOmega )}\big ), \end{aligned}$$
(2.6)
$$\begin{aligned}&\Vert y_u\Vert _{C([0,T];L^2(\varOmega ))} + \Vert y_u\Vert _{L^2(0,T;H_0^1(\varOmega ))}\nonumber \\&\quad \le K\big (\Vert u\Vert _{L^2(Q)} + \Vert a(\cdot ,\cdot ,0)\Vert _{L^2(Q)} + \Vert y_0\Vert _{L^2(\varOmega )}\big ), \end{aligned}$$
(2.7)

for a monotone non-decreasing function \(\eta :[0,\infty ) \longrightarrow [0,\infty )\) and some constant K both independent of u.

Proof

We decompose the state equation into two parts. First, we consider

$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle \frac{\partial z}{\partial t} + Az = u &{} \text{ in } Q,\\ z = 0 \text{ on } \varSigma ,&{} z(0) = y_0 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$
(2.8)

It is well known that it has a unique solution \(z \in W(0,T) \cap L^\infty (Q)\). Moreover, we have the estimates

$$\begin{aligned}&\Vert z\Vert _{W(0,T)} \le C_W(\Vert u\Vert _{L^2(Q)} + \Vert y_0\Vert _{L^2(\varOmega )}), \end{aligned}$$
(2.9)
$$\begin{aligned}&\Vert z\Vert _{L^\infty (Q)} \le C_\infty (\Vert u\Vert _{L^r(0,T;L^p(\varOmega ))} + \Vert y_0\Vert _{L^\infty (\varOmega )}); \end{aligned}$$
(2.10)

see, for instance [17, Chapter III]. Now, we define \(b:Q \times {\mathbb {R}} \longrightarrow {\mathbb {R}}\) as follows

$$\begin{aligned} b(x,t,s) = {{e}}^{-|C_a|t}[a(x,t,{{e}}^{|C_a|t}s + z(x,t)) - a(x,t,z(x,t))] + |C_a|s, \end{aligned}$$

where \(C_a\) is as in (2.2). Then, \(b(x,t,0) = 0\) and according to (2.2)

$$\begin{aligned} \frac{\partial b}{\partial s}(x,t,s) = \frac{\partial a}{\partial s}(x,t,{e}^{|C_a|t}s + z(x,t)) + |C_a| \ge 0. \end{aligned}$$

We consider the equation

$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle \frac{\partial w}{\partial t} + Aw + b(x,t,w) = -{e}^{-|C_a|t}a(x,t,z(x,t)) &{} \text{ in } Q,\\ w = 0 \text{ on } \varSigma , \ \ w(0) = 0 \text{ in } \varOmega .&{}\end{array}\right. \end{aligned}$$
(2.11)

Due to the properties of b, the existence and uniqueness of a solution \(w \in L^\infty (Q) \cap W(0,T)\) is well known; see [20, Theorem 5.5]. Moreover, the following estimates hold

$$\begin{aligned}&\Vert w\Vert _{W(0,T)} \le C_b(\Vert a(\cdot ,\cdot ,z)\Vert _{L^2(Q)} + \Vert b(\cdot ,\cdot ,w)\Vert _{L^2(Q)}), \end{aligned}$$
(2.12)
$$\begin{aligned}&\Vert w\Vert _{L^\infty (Q)} \le C'_\infty \Vert a(\cdot ,\cdot ,z)\Vert _{L^{{\hat{r}}}(0,T;L^{{\hat{p}}}(\varOmega ))}. \end{aligned}$$
(2.13)

Denoting \(M = \Vert z\Vert _{L^\infty (Q)}\) and using (2.4) we infer with the mean value theorem

$$\begin{aligned}&|a(x,t,z(x,t))| \le |a(x,t,z(x,t)) - a(x,t,0)| + |a(x,t,0)|\\&= \Big |\frac{\partial a}{\partial y}(x,t,\theta (x,t)z(x,t))z(x,t)\Big | + |a(x,t,0)| \le C_{a,M}M + |a(x,t,0)|. \end{aligned}$$

Combining this with (2.10) and (2.13) we get

$$\begin{aligned} \Vert w\Vert _{L^\infty (Q)} \le \sigma (\Vert u\Vert _{L^r(0,T;L^p(\varOmega ))} + \Vert a(\cdot ,\cdot ,0)\Vert _{L^{{\hat{r}}}(0,T;L^{{\hat{p}}}(\varOmega ))} + \Vert y_0\Vert _{L^\infty (\varOmega )}\big ),\nonumber \\ \end{aligned}$$
(2.14)

for a non-decreasing function \(\sigma :[0,\infty ) \longrightarrow [0,\infty )\).

If we set \(w = {e}^{-|C_a|t}\psi \) and insert this in (2.11), we infer

$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle \frac{\partial \psi }{\partial t} + A\psi + a(x,t,z(x,t) + \psi ) = 0 &{} \text{ in } Q,\\ \psi = 0 \text{ on } \varSigma ,\ \ \psi (0) = 0 \text{ in } \varOmega .&{}\end{array}\right. \end{aligned}$$
(2.15)

Adding (2.8) and (2.15) we deduce that \(y_u = z + \psi \) solves (1.1). Moreover, any solution of (1.1) is the sum of the solutions of (2.8) and (2.15). Since these equations have a unique solution, the uniqueness of \(y_u\) follows. Furthermore, (2.10) and (2.14) imply (2.6).

To prove (2.7), we take \(\phi = {e}^{-|C_a|t}y_u\) and introduce the function \(f:Q \times {\mathbb {R}} \longrightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} f(x,t,s) = {e}^{-|C_a|t}[a(x,t,{ e}^{|C_a|t}s) - a(x,t,0)] + |C_a|s. \end{aligned}$$

Then, \(\phi \) satisfies

$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle \frac{\partial \phi }{\partial t} + A\phi + f(x,t,\phi ) = { e}^{-|C_a|t}[u -a(x,t,0)] &{} \text{ in } Q,\\ \phi = 0 \text{ on } \varSigma , \ \ \phi (0) = y_0 \text{ in } \varOmega .&{}\end{array}\right. \end{aligned}$$
(2.16)

Since \(f(x,t,0) = 0\) and \(\frac{\partial f}{\partial s}(x,t,s) \ge 0\), multiplying the above equation by \(\phi \), integrating in \(\varOmega \), and using (2.1) we get

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\Vert \phi (t)\Vert ^2_{L^2(\varOmega )} + \varLambda _A\int _\varOmega |\nabla \phi (t)|^2\,\mathrm {d}x\\&\le \frac{1}{2}\frac{d}{dt}\Vert \phi (t)\Vert ^2_{L^2(\varOmega )} + \sum _{i,j = 1}^n\int _\varOmega a_{ij}\partial _{x_i}\phi (t)\partial _{x_j}\phi (t)\,\mathrm {d}x+ \int _\varOmega f(x,t,\phi (t))\phi (t)\,\mathrm {d}x\\&= \int _\varOmega { e}^{-|C_a|t}(u - a(x,t,0))\phi \,\mathrm {d}x\le \big (\Vert u(t)\Vert _{L^2(\varOmega )} + \Vert a(\cdot ,t,0)\Vert _{L^2(\varOmega )}\big )\Vert \phi (t)\Vert _{L^2(\varOmega )}. \end{aligned}$$

Estimate (2.7) follows from this inequality as usual. \(\square \)

We apply Theorem 2.1 with \(p = 2\) and \(r \in \big (\frac{4}{4-n},\infty \big ]\). Observe that \(\frac{1}{r} + \frac{n}{4} < 1\) and \(r > 2\). Then, the mapping \(G:L^r(0,T;L^2(\varOmega )) \longrightarrow L^\infty (Q) \cap W(0,T)\) given by \(G(u) = y_u\) solution of (1.1) is well defined. We have the following differentiability properties of G.

Theorem 2.2

The mapping G is of class \(C^2\). For \(u,v,v_1,v_2 \in L^r(0,T;L^2(\varOmega ))\) the derivatives \(z_v = G'(u)v\) and \(z_{v_1,v_2} = G''(u)(v_1,v_2)\) are the solutions of the equations

$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle \frac{\partial z_v}{\partial t} + A z_v + \frac{\partial a}{\partial y}(x,t,y_u)z_v = v \ \text{ in } Q,\\ z_v = 0 \text{ on } \varSigma ,\ \ z_v(0) = 0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(2.17)
$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle \frac{\partial z_{v_1,v_2}}{\partial t} + A z_{v_1,v_2} + \frac{\partial a}{\partial y}(x,t,y_u)z_{v_1,v_2} + \frac{\partial ^2a}{\partial y^2}(x,t,y_u)z_{v_1}z_{v_2} = 0 \ \text{ in } Q,\\ z_{v_1,v_2} = 0 \text{ on } \varSigma ,\ \ z_{v_1,v_2}(0) = 0 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$
(2.18)

Proof

Let us consider the Banach space

$$\begin{aligned} {\mathcal {Y}} =\{y \in L^\infty (Q) \cap W(0,T) : \frac{\partial y}{\partial t} + Ay \in X\}, \end{aligned}$$

where \(X = L^{{\hat{r}}}(0,T;L^{{\hat{p}}}(\varOmega )) + L^r(0,T;L^2(\varOmega ))\), endowed with the norm

$$\begin{aligned} \Vert y\Vert _{{\mathcal {Y}}} = \Vert y\Vert _{L^\infty (Q)} + \Vert y\Vert _{W(0,T)} + \Vert \frac{\partial y}{\partial t} + Ay\Vert _X. \end{aligned}$$

Now, we define the mapping

$$\begin{aligned}&{\mathcal {F}}:{\mathcal {Y}} \times L^\infty (\varOmega ) \times L^r(0,T;L^2(\varOmega )) \longrightarrow X \times L^\infty (\varOmega )\\&{\mathcal {F}}(y,w,u) =\Big (\frac{\partial y}{\partial t} + A y + a(\cdot ,\cdot ,y) - u, y(0) - w\Big ). \end{aligned}$$

We have that \({\mathcal {F}}\) is of class \(C^2\), \({\mathcal {F}}(y_u,y_0,u) = (0,0)\) for every \(u \in L^r(0,T;L^2(\varOmega ))\), and

$$\begin{aligned}&\frac{\partial {\mathcal {F}}}{\partial y}(y_u,y_0,u):{\mathcal {Y}} \longrightarrow X \times L^\infty (\varOmega )\\&\frac{\partial {\mathcal {F}}}{\partial y}(y_u,y_0,u)z = \Big (\frac{\partial z}{\partial t} + A z + \frac{\partial a}{\partial y}(\cdot ,\cdot ,y_u)z, z(0)\Big ) \end{aligned}$$

is an isomorphism. Hence, an easy application of the implicit function theorem proves the result. \(\square \)

As a consequence of the above theorem and the chain rule we infer the differentiability of the mapping \(J:L^r(0,T;L^2(\varOmega )) \longrightarrow {\mathbb {R}}\). From now on, we assume

$$\begin{aligned} y_d \in L^{{\hat{r}}}(0,T;L^{{\hat{p}}}(\varOmega )), \end{aligned}$$
(2.19)

where \({{\hat{r}}}\) and \({{\hat{p}}}\) are defined in (2.3).

Corollary 2.1

If \(r > \frac{4}{4-n}\), then J is of class \(C^2\) and its derivatives are given by the expressions

$$\begin{aligned}&J'(u)v = \int _Q(\varphi + \kappa u)v\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$
(2.20)
$$\begin{aligned}&J''(u)(v_1,v_2) = \int _Q\Big [\big (1 - \frac{\partial ^2a}{\partial y^2}(x,t,y_u)\varphi \big )z_{v_1}z_{v_2} + \kappa v_1v_2\Big ]\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$
(2.21)

where \(z_{v_i} = G'(u)v_i\), \(i = 1, 2\), and \(\varphi \in C({\bar{Q}}) \cap H^1(Q)\) is the solution of the adjoint state equation

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle -\frac{\partial \varphi }{\partial t} + A^*\varphi + \frac{\partial a}{\partial y}(x,t,y_u)\varphi = y_u - y_d\ \text{ in } Q,\\ \varphi = 0 \text{ on } \varSigma ,\ \ \varphi (T) = 0 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$
(2.22)

Above \(A^*\) denotes the adjoint operator of A

$$\begin{aligned} A^*\varphi =-\sum _{i,j=1}^{n}\partial _{x_j}(a_{ji}(x)\partial _{x_i}\varphi ). \end{aligned}$$

The regularity \({\bar{\varphi }} \in C({\bar{Q}}) \cap H^1(Q)\) follows from Theorems III-6.1 and III-10.1 of [17]. Moreover, we observe that \(J'(u)\) and \(J''(u)\) can be extended to continuous linear and bilinear forms \(J'(u):L^2(Q) \longrightarrow {\mathbb {R}}\) and \(J''(u): L^2(Q) \times L^2(Q) \longrightarrow {\mathbb {R}}\) for every \(u \in L^r(0,T;L^2(\varOmega ))\).

Remark 2.1

Hypotheses (2.1)–(2.5) are satisfied, for instance, for the nonlinearity \(a(y)= \exp (y)\). They are also satisfied for \(a(y)= (y- z_1)(y- z_2)(y- z_3)\) for constants \(z_i\), with \(i\in \{1,2,3\}\). This latter nonlinearity is known in neurology as Nagumo equation and in physical chemistry as Schlögl model. Formulating the optimal control problem with an \(L^1(\varOmega )\) constraint implies that one looks for the action of a controlling laser whose optimal support is small; see [12].

3 Existence of Optimal Controls and First Order Optimality Conditions

Since the control problem (P) is not convex, we need to distinguish between local and global minimizers. We call \({\bar{u}}\) a local minimizer for (P) in the \(L^r(0,T;L^2(\varOmega ))\) sense with \(r > \frac{4}{4-n}\) if \({\bar{u}} \in {U_{ad}}\cap L^\infty (Q)\) and there exists \(\varepsilon > 0\) such that

$$\begin{aligned} J({\bar{u}}) \le J(u)\quad \forall u \in B_\varepsilon \cap {U_{ad}}, \end{aligned}$$
(3.1)

where

$$\begin{aligned} B_\varepsilon =\{u \in L^r(0,T;L^2(\varOmega )) : \Vert u - {\bar{u}}\Vert _{L^r(0,T;L^2(\varOmega ))} \le \varepsilon \}. \end{aligned}$$

It is immediate to check that if \({\bar{u}}\) is a local minimizer in the \(L^r(0,T;L^2(\varOmega ))\) sense, then it is also a local minimizer in the \(L^{r'}(0,T;L^2(\varOmega ))\) sense for every \(r < r' \le \infty \).

Theorem 3.1

There exists at least one solution of (P). Moreover, for every local minimizer \({\bar{u}}\) in the \(L^r(0,T;L^2(\varOmega ))\) sense with \(r > \frac{4}{4-n}\), there exist \({\bar{y}} \in L^2(0,T;H_0^1(\varOmega )) \cap L^\infty (Q)\), \({\bar{\varphi }} \in C({\bar{Q}}) \cap H^1(Q)\), and \({\bar{\mu }} \in L^\infty (Q)\) such that

$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle \frac{\partial {\bar{y}}}{\partial t} + A{\bar{y}} + a(x,t,{\bar{y}}) = {\bar{u}} \ \text{ in } Q,\\ {\bar{y}} = 0 \text{ on } \varSigma ,\ \ {\bar{y}}(0) = y_0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(3.2)
$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle -\frac{\partial {\bar{\varphi }}}{\partial t} + A^*{\bar{\varphi }} + \frac{\partial a}{\partial y}(x,t,{\bar{y}}){\bar{\varphi }} = {\bar{y}} - y_d\ \text{ in } Q,\\ {\bar{\varphi }} = 0 \text{ on } \varSigma ,\ \ {\bar{\varphi }}(T) = 0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(3.3)
$$\begin{aligned}&\int _Q{\bar{\mu }}(u - {\bar{u}})\,\mathrm {d}x\,\mathrm {d}t\le 0\quad \forall u \in {U_{ad}}, \end{aligned}$$
(3.4)
$$\begin{aligned}&{\bar{\varphi }} + \kappa {\bar{u}} + {\bar{\mu }} = 0. \end{aligned}$$
(3.5)

Proof

The proof of existence of a solution for (P) is postponed to the next section, see Theorem 4.5. Given a local minimizer \({\bar{u}}\), we take \({\bar{y}}\) and \({\bar{\varphi }}\) as solutions of (3.2) and (3.3), respectively. Using the convexity of \({U_{ad}}\) and (2.20) we get

$$\begin{aligned} 0 \le J'({\bar{u}})(u - {\bar{u}}) = \int _Q({\bar{\varphi }} + \kappa {\bar{u}})(u - {\bar{u}}) \,\mathrm {d}x\,\mathrm {d}t\quad \forall u \in {U_{ad}}\cap L^\infty (Q). \end{aligned}$$

Now, given \(u \in {U_{ad}}\) arbitrary, we set \(u_k(x,t) = {\text {Proj}}_{[-k,+k]}(u(x,t))\) for \(k \ge 1\), thus \(\{u_k\}_{k= 1}^\infty \subset L^\infty (Q) \cap {U_{ad}}\) and \(u_k \rightarrow u\) in \(L^1(Q)\). Then, we can pass to the limit in the inequality \(J'({\bar{u}})(u_k - {\bar{u}}) \ge 0\) and, hence, we obtain

$$\begin{aligned} \int _Q({\bar{\varphi }} + \kappa {\bar{u}})(u - {\bar{u}}) \,\mathrm {d}x\,\mathrm {d}t\ge 0\ \ \forall u \in {U_{ad}}. \end{aligned}$$

This inequality is equivalent to the fact \(-({\bar{\varphi }} + \kappa {\bar{u}}) \in \partial I_{{U_{ad}}}({\bar{u}}) \subset L^\infty (Q)\). Here \(\partial I_{{U_{ad}}}\) denotes the subdifferential of the indicator function \(I_{{U_{ad}}}:L^1(Q) \longrightarrow [0,+\infty ]\), which takes the value \(I_{U_{ad}}(u) = 0\) if \(u \in {U_{ad}}\) and \(+\infty \) otherwise. Therefore, there exists \({\bar{\mu }} \in \partial I_{{U_{ad}}}\) such that (3.4) and (3.5) holds. \(\square \)

Let us denote by \({\text {Proj}}_{B_\gamma }:L^2(\varOmega ) \longrightarrow B_\gamma \cap L^2(\varOmega )\) the \(L^2(\varOmega )\) projection, where \(B_\gamma = \{v \in L^1(\varOmega ) : \Vert v\Vert _{L^1(\varOmega )} \le \gamma \}\). Then, we have the following consequence of the previous theorem.

Corollary 3.1

Let \({\bar{u}}\), \({\bar{\varphi }}\), and \({\bar{\mu }}\) satisfy (3.2)–(3.5). Then, the following properties hold

$$\begin{aligned}&\int _\varOmega {\bar{\mu }}(t)(v - {\bar{u}}(t))\,\mathrm {d}x\le 0\ \ \forall v \in B_\gamma \text { and for a.a. } t \in (0,T), \end{aligned}$$
(3.6)
$$\begin{aligned}&{\bar{u}}(t) = {\text {Proj}}_{B_\gamma }\big (-\frac{1}{\kappa }{\bar{\varphi }}(t)\big ) \text { for a.a. } t \in (0,T), \end{aligned}$$
(3.7)
$$\begin{aligned}&\left\{ \begin{array}{l}{\bar{u}}(x,t){\bar{\mu }}(x,t) = |{\bar{u}}(x,t)||{\bar{\mu }}(x,t)| \text { for a.a. } (x,t) \in Q,\\ \text {if } \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} < \gamma \text { then } {\bar{\mu }}(t) \equiv 0 \text { in } \varOmega \text { a.e. in } (0,T),\\ \text {if } \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} = \gamma \text { and } {\bar{\mu }}(t) \not \equiv 0 \text { in } \varOmega ,\\ \text { then } {\text {supp}}({\bar{u}}(t)) \subset \{x \in \varOmega : |{\bar{\mu }}(x,t)| = \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\}.\end{array}\right. \end{aligned}$$
(3.8)

Proof

Let us show that (3.4) and (3.6) are equivalent. Using Fubini’s theorem, it is obvious that (3.6) implies (3.4). Let us prove the contrary implication. Let \(v \in B_\gamma \) be arbitrary and set

$$\begin{aligned} I_v = \Big \{t \in (0,T) : \int _\varOmega {\bar{\mu }}(x,t)(v(x) - {\bar{u}}(x,t))\,\mathrm {d}x> 0\Big \} \end{aligned}$$

and

$$\begin{aligned} u(x,t) = \left\{ \begin{array}{cl}v(x) &{}\text {if } t \in I_v,\\ {\bar{u}}(x,t)&{}\text {otherwise.}\end{array}\right. \end{aligned}$$

Then, \(u \in {U_{ad}}\) and (3.4) yields

$$\begin{aligned} 0 \ge \int _Q{\bar{\mu }}(x,t)(u - {\bar{u}})\,\mathrm {d}x\,\mathrm {d}t= \int _{I_v}\int _\varOmega {\bar{\mu }}(x,t)(v(x) - {\bar{u}}(x,t))\,\mathrm {d}x. \end{aligned}$$

This is only possible if \(|I_v| = 0\). In order to prove (3.7) we use (3.5) and (3.6) to get

$$\begin{aligned} \int _\varOmega \Big (-\frac{1}{\kappa }{\bar{\varphi }}(t) - {\bar{u}}(t)\Big )(v - {\bar{u}}(t)) \le 0 \ \ \forall v \in B_\gamma \cap L^2(\varOmega ) \text { and for a.a. } t \in (0,T). \end{aligned}$$

Since \(B_\gamma \cap L^2(\varOmega )\) is a convex and closed subset of \(L^2(\varOmega )\), the above inequality is the well known characterization of (3.7).

Let us prove the first statement of (3.8). Take \(u(x,t) = {\text {sign}}({\bar{\mu }}(x,t))|{\bar{u}}(x,t)|\). Then, \(u \in {U_{ad}}\) and with (3.4) we obtain

$$\begin{aligned} \int _Q|{\bar{\mu }}(x,t)||{\bar{u}}(x,t)|\,\mathrm {d}x\,\mathrm {d}t= \int _Q{\bar{\mu }}(x,t)u(x,t)\,\mathrm {d}x\,\mathrm {d}t\le \int _Q{\bar{\mu }}(x,t){\bar{u}}(x,t)\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$

which proves the desired identity. We prove the second statement of (3.8). For every \(\varepsilon > 0\) we define

$$\begin{aligned} I_\varepsilon = \{t \in (0,T) : \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} \le \gamma - \varepsilon \}. \end{aligned}$$

Denote \(B_\varepsilon \) the closed ball of \(L^1(\varOmega )\) centered at 0 and radius \(\varepsilon \). Take \(v \in B_\varepsilon \) arbitrary. Then, we have that \(v+{\bar{u}}(t) \in B_\gamma \) for \(t \in I_\varepsilon \), and (3.6) yields

$$\begin{aligned} \int _\varOmega {\bar{\mu }}(x,t)v(x)\,\mathrm {d}x\le 0\ \ \forall v \in B_\varepsilon \text { and } t \in I_\varepsilon , \end{aligned}$$

which implies that \({\bar{\mu }}(t) \equiv 0\) in \(\varOmega \) for \(t \in I_\varepsilon \). Since \(\varepsilon > 0\) is arbitrary, we infer the second statement of (3.8). Let us prove the third statement. Under the assumption \(\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} = \gamma \) and \({\bar{\mu }}(t) \not \equiv 0\) in \(\varOmega \). For every \(\varepsilon > 0\) and \(t \in (0,T)\) we consider the sets

$$\begin{aligned}&\varOmega ^\varepsilon (t) = \{x \in \varOmega : |{\bar{u}}(x,t)|> \varepsilon \text { and } |{\bar{\mu }}(x,t)| < \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )} - \varepsilon \},\\&{\tilde{\varOmega }}^\varepsilon (t) = \{x \in \varOmega : |{\bar{\mu }}(x,t)| > \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )} - \varepsilon \}. \end{aligned}$$

We are going to prove that \(|\varOmega ^\varepsilon (t)| = 0\) for almost all \(t \in (0,T)\). Assume that \(|\varOmega ^\varepsilon (t)| > 0\) for some \(\varepsilon > 0\) and \(t \in (0,T)\). Since \(|{\tilde{\varOmega }}^\varepsilon (t)| > 0\) by definition of the essential supremum, we can find two sets \(E \subset \varOmega ^\varepsilon (t)\) and \(F \subset {\tilde{\varOmega }}^\varepsilon (t)\) such that \(|E| = |F| > 0\). We define the control

$$\begin{aligned} v(x) = \left\{ \begin{array}{cl}{\bar{u}}(x,t) - \varepsilon {\text {sign}}({\bar{u}}(x,t))&{}\text {if } x \in E,\\ {\bar{u}}(x,t) + \varepsilon {\text {sign}}({\bar{u}}(x,t))&{}\text {if } x \in F,\\ {\bar{u}}(x,t)&{}\text {otherwise.}\end{array}\right. \end{aligned}$$

Since \(\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} = \gamma \), we get

$$\begin{aligned} \Vert v\Vert _{L^1(\varOmega )} = \int _E|{\bar{u}}(x)|\,\mathrm {d}x- \varepsilon |E| + \int _F|{\bar{u}}(x)|\,\mathrm {d}x+ \varepsilon |F| + \int _{\varOmega \setminus (E\cup F)}|{\bar{u}}(x)|\,\mathrm {d}x= \gamma . \end{aligned}$$

Moreover, we get with the first statement of (3.8)

$$\begin{aligned} \int _\varOmega {\bar{\mu }}(x,t)(v(x)- {\bar{u}}(x,t))\,\mathrm {d}x= -\varepsilon \int _E|{\bar{\mu }}(x,t)|\,\mathrm {d}x+ \varepsilon \int _F|{\bar{\mu }}(x,t)|\,\mathrm {d}x> 0, \end{aligned}$$

which contradicts (3.6) unless it is satisfied for a set of points t of zero Lebesgue measure. Taking

$$\begin{aligned} \varOmega (t) = \{x \in \varOmega : |{\bar{u}}(x,t)| > 0 \text { and } |{\bar{\mu }}(x,t)| < \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\}, \end{aligned}$$

since \(\varepsilon > 0\) was arbitrary, we deduce that \(|\varOmega (t)| = 0\) for almost all \(t \in (0,T)\). This implies that \({\text {supp}}({\bar{u}}(t)) \subset \{x \in \varOmega : |{\bar{\mu }}(x,t)| = \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\}\). \(\square \)

Remark 3.1

Let us observe that the first statement of (3.8) and (3.5) imply

$$\begin{aligned} |{\bar{\varphi }}(x,t)| = \kappa |{\bar{u}}(x,t)| + |{\bar{\mu }}(x,t)|. \end{aligned}$$

This yields

$$\begin{aligned} \Vert {\bar{\varphi }}(t)\Vert _{L^1(\varOmega )} = \kappa \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} + \Vert {\bar{\mu }}(t)\Vert _{L^1(\varOmega )}. \end{aligned}$$

From this identity and the second statement of (3.8) we infer that \({\bar{\mu }}(t) \not \equiv 0\) in \(\varOmega \) if and only if \(\Vert {\bar{\varphi }}(t)\Vert _{L^1(\varOmega )} > \kappa \gamma \).

Remark 3.2

From (3.8) we deduce that \({\bar{\mu }}(x,t) \in \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\,\partial |\cdot |({\bar{u}}(x,t))\) for almost every point \((x,t) \in Q\).

Corollary 3.2

Let \({\bar{u}} \in {U_{ad}}\cap L^\infty (\varOmega )\) satisfy (3.5) and (3.8). Then, the following identities are satisfied

$$\begin{aligned}&{\bar{u}}(x,t) = -\frac{1}{\kappa }{\text {sign}}({\bar{\varphi }}(x,t))\big (|{\bar{\varphi }}(x,t)| - \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\big )^+\nonumber \\&=-\frac{1}{\kappa }\left\{ \Big [{\bar{\varphi }}(x,t) + \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\Big ]^- + \Big [{\bar{\varphi }}(x,t) - \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\Big ]^+\right\} . \end{aligned}$$
(3.9)

Moreover, the regularity \({\bar{u}} \in H^1(Q)\) and \({\bar{\mu }} \in H^1(Q)\) hold.

Proof

If \(\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )} = 0\), then \({\bar{u}}(x,t) = -\frac{1}{\kappa }{\bar{\varphi }}(x,t)\) follows from (3.5), which coincides with the identity (3.9). Now, we assume that \(\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )} > 0\). Using (3.8) we obtain that \(\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} = \gamma \). Then, the third statement of (3.8) implies that \(|{\bar{\mu }}(x,t)| = \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\) if \(|{\bar{u}}(x,t)| > 0\). We distinguish three cases.

  1. (i)

    If \({\bar{u}}(x,t) > 0\), (3.5) and the first statement of (3.8) leads to \({\bar{u}}(x,t) = -\frac{1}{\kappa }({\bar{\varphi }}(x,t) + \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )})\), which coincides with the expression (3.9).

  2. (ii)

    If \({\bar{u}}(x,t) = 0\), using again (3.5) we get \(|{\bar{\varphi }}(x,t)| = |{\bar{\mu }}(x,t)| \le \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\). Then, the identity (3.9) holds.

  3. (iii)

    If \({\bar{u}}(x,t) < 0\), from the first statement of (3.8) and (3.5) we infer that \({\bar{u}}(x,t) = -\frac{1}{\kappa }({\bar{\varphi }}(x,t) - \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )})\). Then, (3.9) holds too. The spatial regularity \({\bar{u}} \in L^2(0,T;H_0^1(\varOmega ))\) is an immediate consequence of (3.9) and the fact that \({\bar{\varphi }} \in H^1(Q)\). For the temporal regularity of \({\bar{u}}\), we first observe

    $$\begin{aligned}&\Vert {\bar{u}}(t) - {\bar{u}}(t')\Vert _{L^2(\varOmega )}\\&= \Vert {\text {Proj}}_{B_\gamma }(-\frac{1}{\kappa }{\bar{\varphi }}(t)) - {\text {Proj}}_{B_\gamma }(-\frac{1}{\kappa }{\bar{\varphi }}(t'))\Vert _{L^2(\varOmega )} \le \frac{1}{\kappa }\Vert {\bar{\varphi }}(t) - {\bar{\varphi }}(t')\Vert _{L^2(\varOmega )}. \end{aligned}$$

    Since \({\bar{\varphi }}:[0,T] \longrightarrow L^2(\varOmega )\) is absolutely continuous, using the above inequality we infer that \({\bar{u}}:[0,T] \longrightarrow L^2(\varOmega )\) is also absolutely continuous. Moreover, the same inequality yields \(\Vert {\bar{u}}'(t)\Vert _{L^2(\varOmega )} \le \frac{1}{\kappa }\Vert {\bar{\varphi }}'(t)\Vert _{L^2(\varOmega )}\) and \({\bar{u}} \in W^{1,2}(0,T;L^2(\varOmega ))\). All together, this implies that \({\bar{u}} \in H^1(Q)\). The regularity of \({\bar{\mu }}\) follows from (3.5). \(\square \)

Corollary 3.3

Let \({\bar{u}}\) be as in Corollary 3.2. Then, we have the following property

$$\begin{aligned} {\bar{u}}(x,t) = 0 \text { if and only if } |{\bar{\varphi }}(x,t)| \le \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}. \end{aligned}$$
(3.10)

This corollary is a straightforward consequence of (3.9).

Theorem 3.2

There exists a constant \(K_\infty > 0\) independent of \(\gamma \) such that \(\Vert {\bar{u}}\Vert _{L^\infty (Q)} \le K_\infty \) for every global minimizer \({\bar{u}}\) of (P). In addition, if we set \(\gamma _0 = K_\infty |\varOmega |\), then for every \(\gamma > \gamma _0\) and every solution \({\bar{u}}\) of (P) we have \(\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} < \gamma \) for almost every t and \({\bar{u}} = -\frac{1}{\kappa }{\bar{\varphi }}\).

To prove this theorem, we can argue as in the proof of Theorem 4.4 below to deduce the existence of \(K_\infty > 0\) independent of \(\gamma \) such that \(\Vert {\bar{u}}\Vert _{L^\infty (Q)} \le K_\infty \). The last statement is a straightforward consequence of this estimate and the definition of \(\gamma _0\).

4 Proof of Existence of a Solution for (P)

The proof of existence of a solution of (P) can not be performed by the classical method of calculus of variations due to the lack of boundedness of \({U_{ad}}\) in \(L^\infty (\varOmega )\) and the non coercivity of J on this space. One can try to prove the existence of a solution \({\bar{u}}\) of (P) in \(L^2(Q)\) and then to deduce that \({\bar{u}} \in L^\infty (Q)\) from the optimality conditions. However, the differentiability of J in \(L^2(Q)\) can fail due to the nonlinearity of the state equation. To overcome this difficulty we are going to truncate the nonlinear term a(xty) as follows. For every \(M > 0\) we define the function \(f_M:{\mathbb {R}} \longrightarrow {\mathbb {R}}\) by

$$\begin{aligned} f_M(s) = \left\{ \begin{array}{cl} M + 1&{} \text {if } s > M + 1,\\ s+(M - s)^2 + (M - s)^3&{} \text {if } M \le s \le M + 1,\\ s&{}\text {if } -M< s< +M,\\ s - (M + s)^2 - (M+s)^3&{}\text {if } -M - 1 \le s \le -M,\\ -M - 1&{}\text {if } s < -M - 1. \end{array}\right. \end{aligned}$$

It can be easily checked that \(f_M \in C^1({\mathbb {R}})\) and \(0 \le f'_M(s) \le 1\) for every \(s \in {\mathbb {R}}\). Now, we set \(a_M(x,t,s) = a(x,t,f_M(s))\). It is obvious that \(a_M\) is of class \(C^1\) with respect to the last variable and (2.2)–(2.4) imply

$$\begin{aligned}&\frac{\partial a_M}{\partial y}(x,t,y) = \frac{\partial a}{\partial y}(x,t,f_M(y))f'_M(y)\ge \min (0,C_a)\ \forall y\in {\mathbb {R}}, \end{aligned}$$
(4.1)
$$\begin{aligned}&a_M(\cdot ,\cdot ,0) = a(\cdot ,\cdot ,0)\in L^{{{\hat{r}}}}(0,T;L^{{{\hat{p}}}}(\varOmega )), \end{aligned}$$
(4.2)
$$\begin{aligned}&\left| \frac{\partial a_M}{\partial y}(x,t,y)\right| \le C_{a,M+1}\ \forall y \in {\mathbb {R}}, \end{aligned}$$
(4.3)

for almost all \((x,t) \in Q\).

Theorem 4.1

For any \(M > 0\) and all \(u \in L^2(Q)\) the equation

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \frac{\partial y}{\partial t} + A y + a_M(x,t,y) = u \ \text{ in } Q,\\ y = 0 \text{ on } \varSigma ,\ \ y(0) = y_0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(4.4)

has a unique solution \(y^M_u \in W(0,T)\). Moreover, \(y^M_u\) satisfies the inequalities

$$\begin{aligned}&\Vert y^M_u\Vert _{C(0,T;L^2(\varOmega ))} + \Vert y^M_u\Vert _{L^2(0,T;H_0^1(\varOmega ))}\nonumber \\&\le K\big (\Vert u\Vert _{L^2(Q)} + \Vert a(\cdot ,\cdot ,0)\Vert _{L^2(Q)} + \Vert y_0\Vert _{L^2(\varOmega )}\big ), \end{aligned}$$
(4.5)
$$\begin{aligned}&\Vert y^M_u\Vert _{W(0,T)}\nonumber \\&\le K'\big (\Vert u\Vert _{L^2(Q)} + \Vert y_0\Vert _{L^2(\varOmega )} + \Vert a(\cdot ,\cdot ,0)\Vert _{L^2(Q)} + C_{a,M+1}(M+1)|Q|^{\frac{1}{2}}\big ), \end{aligned}$$
(4.6)

where K is the same constant as in (2.7) and \(K'\) is independent of M and u.

Proof

From (4.3) and the mean value theorem we infer that \(|a_M(\cdot ,\cdot ,s) - a_M(\cdot ,\cdot ,0)| \le C_{a,M+1}(M+1)\) for all \(s \in {\mathbb {R}}\). Consequently, the estimate

$$\begin{aligned} \Vert a_M(\cdot ,\cdot ,y) - a_M(\cdot ,\cdot ,0)\Vert _{L^2(Q)} \le C_{a,M+1}(M+1)|Q|^{\frac{1}{2}} \end{aligned}$$

holds. Hence, an easy application of fixed point Schauder’s theorem yields the existence of a solution \(y^M_u\) in W(0, T). The uniqueness follows in the standard way noting that

$$\begin{aligned} \int _\varOmega [a_M(x,t,y_2) - a_M(x,t,y_1)](y_2 - y_1)\,\mathrm {d}x\ge \min \{0,C_a\}\Vert y_2 - y_1\Vert ^2_{L^2(\varOmega )}. \end{aligned}$$

The proof of the estimate (4.5) is the same as the one of (2.7). Inequality (4.6) follows from (4.5) and the fact that

$$\begin{aligned} \Vert a_M(\cdot ,\cdot ,y)\Vert _{L^2(Q)} \le \Vert a_M(\cdot ,\cdot ,0)\Vert _{L^2(Q)} + C_{a,M+1}(M+1)|Q|^{\frac{1}{2}}. \end{aligned}$$

\(\square \)

Let us define the mapping \(G_M:L^2(Q) \longrightarrow W(0,T)\) associating to every u the corresponding solution \(y^M_u\) of (4.4).

Theorem 4.2

The mapping \(G_M\) is of class \(C^1\). For all \(u, v \in L^2(Q)\) the derivative \(z_v = G'_M(u)v\) is the solution of the linearized equation

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \frac{\partial z}{\partial t} + A z + \frac{\partial a_M}{\partial y}(x,t,y^M_u)z = v \ \text{ in } Q,\\ z = 0 \text{ on } \varSigma ,\ \ z(0) = 0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(4.7)

where \(y^M_u = G_M(u)\).

Proof

Let us introduce the space

$$\begin{aligned} Y = \Big \{y \in W(0,T) : \frac{\partial y}{\partial t} + A y \in L^2(Q)\Big \}. \end{aligned}$$

This is a Banach space when it is endowed with the graph norm

$$\begin{aligned} \Vert y\Vert _Y = \Vert y\Vert _{W(0,T)} + \Vert \frac{\partial y}{\partial t} + A y\Vert _{L^2(Q)}. \end{aligned}$$

Now, we define the mapping

$$\begin{aligned}&{\mathcal {F}}_M:Y \times L^2(\varOmega ) \times L^2(Q) \longrightarrow L^2(Q) \times L^2(\varOmega )\\&{\mathcal {F}}_M(y,w,u) = \Big (\frac{\partial y}{\partial t} + A y + a_M(\cdot ,\cdot ,y) - u, y(0) - w\Big ). \end{aligned}$$

Let us prove that the mapping

$$\begin{aligned} F_M:W(0,T) \longrightarrow L^2(Q),\quad F_M(y) = a_M(\cdot ,\cdot ,y) \end{aligned}$$

is of class \(C^1\) with

$$\begin{aligned} DF_M:W(0,T) \longrightarrow {\mathcal {L}}(W(0,T),L^2(Q)),\quad DF_M(y)z= \frac{\partial a_M}{\partial s}(\cdot ,\cdot ,y)z. \end{aligned}$$

First, we observe that a standard application of a Gagliardo–Nirenberg inequality leads to

$$\begin{aligned} \Vert z\Vert _{L^{\frac{8}{3}}(0,T;L^4(\varOmega ))} \le C\Vert z\Vert ^{\frac{1}{4}}_{L^\infty (0,T;L^2(\varOmega ))}\Vert z\Vert ^{\frac{3}{4}}_{L^2(0,T;H_0^1(\varOmega ))} \le C'\Vert z\Vert _{W(0,T)} \end{aligned}$$

for every \(z \in W(0,T)\). Using this inequality, (4.3), and the mean value theorem we infer

$$\begin{aligned}&\Vert F_M(y + z) - F_M(y) - DF_M(y)z\Vert ^2_{L^2(Q)}\\&\quad =\int _Q\Big |a_M(x,t,y(x,t)+z(x,t)) - a_M(x,t,y(x,t)) - \frac{\partial a_M}{\partial s}(x,t,y(x,t))z(x,t)\Big |^2 \,\mathrm {d}x\,\mathrm {d}t\\&\quad =\int _Q\Big |\frac{\partial a_M}{\partial s}(x,t,y(x,t)+\theta (x,t)z(x,t)) - \frac{\partial a_M}{\partial s}(x,t,y(x,t))\Big |^2z^2(x,t)\,\mathrm {d}x\,\mathrm {d}t\\&\quad \le \int _0^T\Big \Vert \frac{\partial a_M}{\partial s}(\cdot ,t,y(t)+\theta (t)z(t)) - \frac{\partial a_M}{\partial s}(\cdot ,t,y(t))\Big \Vert _{L^4(\varOmega )}^2\Vert z(t)\Vert ^2_{L^4(\varOmega )}\,\mathrm {d}t\\&\quad \le \Big \Vert \frac{\partial a_M}{\partial s}(\cdot ,\cdot ,y+\theta z) - \frac{\partial a_M}{\partial s}(\cdot ,\cdot ,y)\Big \Vert ^2_{L^8(0,T;L^4(\varOmega ))}\Vert z\Vert ^2_{L^{\frac{8}{3}}(0,T;L^4(\varOmega ))}. \end{aligned}$$

From here we deduce

$$\begin{aligned} \lim _{\Vert z\Vert _{W(0,T)} \rightarrow 0}\frac{\Vert F_M(y + z) - F_M(y) - DF_M(y)z\Vert _{L^2(Q)}}{\Vert z\Vert _{W(0,T)}} = 0. \end{aligned}$$

Hence, \(F_M\) is Fréchet differentiable. The continuity of \(DF_M\) is immediate and, consequently, \(F_M\) is of class \(C^1\). Using this and the continuity of the embedding \(Y \subset W(0,T) \subset C([0,T];L^2(\varOmega ))\), we conclude that \({\mathcal {F}}_M\) is of class \(C^1\). Moreover, we have \({\mathcal {F}}_M(y^M_u,y_0,u) = (0,0)\). An easy application of the implicit function theorem proves Theorem 4.2. \(\square \)

For every \(M > 0\) we consider the control problems

$$\begin{aligned} \text{(P }_M) \quad \inf _{u \in {U_{ad}}\cap L^2(Q)} J_M(u):= & {} \frac{1}{2}\int _Q (y^M_u(x,t) - y_d(x,t))^2\,\mathrm {d}x\,\mathrm {d}t\\&+ \frac{\kappa }{2}\int _Q u(x,t)^2\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$

where \(y^M_u\) denotes the solution of (4.4). Problem \( \text{(P }_M) \) has at least a solution \(u_M\). This is consequence of the coercivity of \(J_M\) on \(L^2(Q)\), the fact that \({U_{ad}}\cap L^2(Q)\) is closed and convex in \(L^2(Q)\), and the lower semicontinuity of \(J_M\) with respect to the weak topology of \(L^2(Q)\). The last statement follows easily from the estimate (4.6) and the compactness of the embedding \(W(0,T) \subset L^2(Q)\).

From the chain rule and Theorem 4.2 we infer that \(J_M : L^2(Q) \longrightarrow {\mathbb {R}}\) is of class \(C^1\) and its derivative is given by the expression

$$\begin{aligned} J'_M(u)v = \int _Q(\varphi + \kappa u)v\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$
(4.8)

where \(\varphi \in W(0,T)\) is the solution of the adjoint state equation

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle -\frac{\partial \varphi }{\partial t} + A^*\varphi + \frac{\partial a_M}{\partial y}(x,t,y^M_u)\varphi = y^M_u - y_d\ \text{ in } Q,\\ \varphi = 0 \text{ on } \varSigma ,\ \ \varphi (T) = 0 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$
(4.9)

Theorem 4.3

Let \(u_M\) be a solution of \( \text{(P }_M) \). Then, there exist functions \(y_M, \varphi _M \in W(0,T)\) and \(\mu _M \in L^2(Q)\) such that

$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle \frac{\partial y_M}{\partial t} + A y_M + a_M(x,t,y_M) = u_M \ \text{ in } Q,\\ y_M = 0 \text{ on } \varSigma ,\ \ y_M(0) = y_0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(4.10)
$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle -\frac{\partial \varphi _M}{\partial t} + A^*\varphi _M + \frac{\partial a_M}{\partial y}(x,t,y_M)\varphi _M = y_M - y_d\ \text{ in } Q,\\ \varphi _M = 0 \text{ on } \varSigma ,\ \ \varphi _M(T) = 0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(4.11)
$$\begin{aligned}&\int _Q\mu _M(x,t)(u(x,t) - u_M(x,t))\,\mathrm {d}x\,\mathrm {d}t\le 0\quad \forall u \in {U_{ad}}\cap L^2(Q),\end{aligned}$$
(4.12)
$$\begin{aligned}&\varphi _M + \kappa u_M + \mu _M = 0. \end{aligned}$$
(4.13)

The proof of this theorem is the same as the one of Theorem 3.1.

Theorem 4.4

Let \((u_M,y_M,\varphi _M,\mu _M)\) be as in Theorem 4.3. Then, there exists a constant \(K_\infty > 0\) such that

$$\begin{aligned} \Vert (u_M,y_M,\varphi _M,\mu _M)\Vert _{L^\infty (Q)^4} \le K_\infty \quad \forall M > 0. \end{aligned}$$
(4.14)

Proof

As in the proof for the first statement of (3.8), we have that (4.12) and (4.13) yield \(|\mu _M(x,t)||u_M(x,t)| = \mu _M(x,t)u_M(x,t)\) for almost all \((x,t) \in Q\).

We denote by \(y_M^0\) the solution of (4.4) associated with the control identically zero. Then, according to Theorem 4.1, inequality (4.5) implies that

$$\begin{aligned} \Vert y_M^0\Vert _{C(0,T;L^2(\varOmega ))} \le K\big (\Vert a(\cdot ,\cdot ,0)\Vert _{L^2(Q)} + \Vert y_0\Vert _{L^2(\varOmega )}\big )\quad \forall M > 0. \end{aligned}$$

From this inequality we infer

$$\begin{aligned} \Vert y^0_M\Vert _{L^2(Q)} \le C_1 = \sqrt{T}K\big (\Vert a(\cdot ,\cdot ,0)\Vert _{L^2(Q)} + \Vert y_0\Vert _{L^2(\varOmega )}\big ) \quad \forall M > 0. \end{aligned}$$

Since \(u_M\) is solution of \( \text{(P }_M) \) and \(u \equiv 0\) is an admissible control for \( \text{(P }_M) \) we get

$$\begin{aligned} \frac{\kappa }{2}\Vert u_M\Vert ^2_{L^2(Q)} \le J_M(u_M) \le J_M(0) = \frac{1}{2}\Vert y^0_M - y_d\Vert ^2_{L^2(Q)}. \end{aligned}$$

This leads to

$$\begin{aligned} \Vert u_M\Vert _{L^2(Q)} \le \frac{1}{\sqrt{\kappa }}\Vert y^0_M - y_d\Vert _{L^2(Q)} \le C_2 = \frac{1}{\sqrt{\kappa }}(C_1 + \Vert y_d\Vert _{L^2(Q)}) \quad \forall M > 0. \end{aligned}$$

Using again (4.5) and this estimate we deduce

$$\begin{aligned} \Vert y_M\Vert _{L^\infty (0,T;L^2(\varOmega ))} \le C_3= & {} K_2\big (C_2 + \Vert a(\cdot ,\cdot ,0)\Vert _{L^2(Q)} \\&+ \Vert b(\cdot ,\cdot ,0)\Vert _{L^2(\varSigma )} + \Vert y_0\Vert _{L^2(\varOmega )}\big ) \quad \forall M > 0. \end{aligned}$$

Using this estimate we can infer the boundedness of \(\varphi _M\) by a constant independent of M. The idea of the proof is to make the substitution \(\varphi _M(x,t) = \mathrm{e}^{-|C_a|t}\psi _M(x,t)\), where \(C_a\) is given in (2.2). Then, \(\psi \) satisfies the equation

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle -\frac{\partial \psi _M}{\partial t} + A^*\psi _M + \big (\frac{\partial a_M}{\partial y}(x,t,y_M) + |C_a|\big )\psi _M = \mathrm{e}^{|C_a|t}(y_M - y_d)\ \text{ in } Q,\\ \psi _M = 0 \text{ on } \varSigma ,\ \ \psi _M(T) = 0 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$

Since (4.1) implies that \(\frac{\partial a_M}{\partial y}(x,t,y^M_u) + |C_a| \ge 0\), we apply [17, Theorem III-7.1] to deduce the existence of a constant \(C >0 \) independent of M such that

$$\begin{aligned}&\Vert \psi _M\Vert _{L^\infty (Q)} \le C\big (\mathrm{e}^{|C_a|T}\big [\Vert y_M\Vert _{L^\infty (0,T;L^2(\varOmega ))} + \Vert y_d\Vert _{L^{{\hat{r}}}(0,T;L^{{\hat{p}}}(\varOmega ))}\big ]\big )\\&\le C_4 = C\big (\mathrm{e}^{|C_a|T}\big [C_3 + \Vert y_d\Vert _{L^{{\hat{r}}}(0,T;L^{{\hat{p}}}(\varOmega ))}\big ]\big )\quad \forall M > 0. \end{aligned}$$

From here we infer the estimate \(\Vert \varphi _M\Vert _{L^\infty (Q)} \le \Vert \psi _M\Vert _{L^\infty (Q)} \le C_4\) for every \(M > 0\). Now, using that \(u_M\) and \(\mu _M\) have the same sign almost everywhere in Q, we deduce from (4.13)

$$\begin{aligned} \kappa |u_M(x,t)| \le |\kappa u_M(x,t) + \mu _M(x,t)| = |\varphi _M(x,t)| \le C_4, \end{aligned}$$

which proves that \(\Vert u_M\Vert _{L^\infty (Q)} \le \frac{C_4}{\kappa }\) for every \(M > 0\). Moreover, the bounds from \(u_M\) and \(\varphi _M\) along with (4.13) imply that \(\Vert \mu _M\Vert _{L^\infty (Q)} \le 2C_4\). Finally, the estimate of \(y_M\) in \(L^\infty (Q)\) independently of M follows from (4.10), Theorem 2.1, and the estimate for \(u_M\). \(\square \)

Remark 4.1

The assumption \(\kappa > 0\) was used in an essential manner in the above proof.

Theorem 4.5

Let \(M \ge K_\infty \) be arbitrary, where \(K_\infty \) satisfies (4.14). Let \(u_M\) be a solution of \( \text{(P }_M) \). Then, \(u_M\) is a solution of (P).

Proof

First we observe that \(\Vert y_M\Vert _{L^\infty (Q)} \le M\) and hence \(a_M(x,t,y_M) = a(x,t,y_M)\). Therefore, \(y_M\) is the solution of (1.1) corresponding to \(u_M\) and, consequently, \(J_M(u_M) = J(u_M)\).

Given \(u \in {U_{ad}}\cap L^\infty (Q)\) arbitrary, let \(y_u\) be the associated solution of (1.1) and set \(M_0 = \Vert y_u\Vert _{L^\infty (Q)}\). If \(M_0 \le M\), then it is obvious that \(a_M(x,t,y_u) = a(x,t,y_u)\) and, hence, \(J_M(u) = J(u)\). Therefore, the optimality of \(u_M\) implies \(J(u_M) = J_M(u_M) \le J_M(u) = J(u)\).

If \(M_0 > M\), we take a solution \(u_{M_0}\) of (P\(_{M_0}\)). Then, Theorem 4.4 implies that the solution \(y_{M_0}\) of (4.10) with M replaced by \(M_0\) satisfies \(\Vert y_{M_0}\Vert _{L^\infty (Q)} \le M\) and, consequently, \(a_{M_0}(x,t,y_{M_0}) = a_M(x,t,y_{M_0}) = a(x,t,y_{M_0})\) and \(J_{M_0}(u_{M_0}) = J_M(u_{M_0}) = J(u_{M_0})\). These facts along with the optimality of \(u_M\) and \(u_{M_0}\) lead to

$$\begin{aligned} J(u_M) = J_M(u_M) \le J_M(u_{M_0}) = J_{M_0}(u_{M_0}) \le J_{M_0}(u) = J(u), \end{aligned}$$

which proves that \(u_M\) is a solution of (P). \(\square \)

Remark 4.2

Let us compare problem (P) with the control problems

$$\begin{aligned} \text{(P }_r) \quad \inf _{u \in {U_{ad}}\cap L^r(0,T;L^2(\varOmega ))} J(u):= & {} \frac{1}{2}\int _Q (y_u(x,t) - y_d(x))^2\,\mathrm {d}x\,\mathrm {d}t\\&+ \frac{\kappa }{2}\int _Q u(x,t)^2\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$

where \(r \in (\frac{4}{4 - n},\infty )\). We observe that Theorems 2.1 and 2.2 , and Corollary 2.1 are applicable to deduce that any solution of \( \text{(P }_r) \) satisfies the optimality conditions (3.2)–(3.5). Then, the arguments of Theorem 4.4 apply to deduce that any solution of \( \text{(P }_r) \) belongs to \(L^\infty (Q)\). Let us check that problems (P) and \( \text{(P }_r) \) are equivalent in the sense that both have the same solutions. Indeed, since \({U_{ad}}\cap L^r(0,T;L^2(\varOmega )) \supset {U_{ad}}\cap L^\infty (Q)\), it is obvious that every solution of \( \text{(P }_r) \) is a solution of (P). Conversely, let \({\bar{u}}\) be a solution of (P) and take \(u \in {U_{ad}}\cap L^r(0,T;L^2(\varOmega ))\) arbitrarily. For every integer \(k \ge 1\) we set \(u_k = {\text {Proj}}_{[-k,+k]}(u)\). Then, it is obvious that \(u_k \in {U_{ad}}\cap L^\infty (Q)\) and \(u_k \rightarrow u\) in \(L^r(0,T;L^2(\varOmega ))\). Using the optimality of \({\bar{u}}\) we have \(J({\bar{u}}) \le J(u_k)\) for all k, and passing to the limit we infer that \(J({\bar{u}}) \le J(u)\). Since u was arbitrary, this implies that \({\bar{u}}\) is a solution of \( \text{(P }_r) \).

5 Second Order Optimality Conditions

We consider the Lipschitz and convex mapping \(j:L^1(\varOmega ) \longrightarrow {\mathbb {R}}\) defined by \(j(v) = \Vert v\Vert _{L^1(\varOmega )}\). Its directional derivative is given by the expression

$$\begin{aligned} j'(u;v) = \int _{\varOmega ^+_u}v(x)\,\mathrm {d}x- \int _{\varOmega ^-_u}v(x)\,\mathrm {d}x+ \int _{\varOmega ^0_u}|v(x)|\,\mathrm {d}x\quad \forall u, v \in L^1(\varOmega ), \end{aligned}$$
(5.1)

where

$$\begin{aligned} \varOmega ^+_u =\{x \in \varOmega : u(x) > 0\}, \ \varOmega ^-_u =\{x \in \varOmega : u(x) < 0\} \text { and } \varOmega _u^0 = \varOmega \setminus (\varOmega _u^+ \cup \varOmega _u^-). \end{aligned}$$

In order to derive the second order optimality conditions for (P), we define the cone of critical directions. For a control \({\bar{u}} \in {U_{ad}}\cap L^\infty (Q)\) satisfying the first order optimality conditions (3.2)–(3.5) we set

$$\begin{aligned} C_{{\bar{u}}} = \Big \{v \in L^2(Q) : J'({\bar{u}})v = 0 \text { and } j'({\bar{u}}(t);v(t)) \left\{ \begin{array}{cl} = 0&{}\text {if } t \in I_\gamma ^+,\\ \le 0&{}\text {if } t \in I_\gamma \setminus I^+_\gamma ,\end{array}\right. \Big \}, \end{aligned}$$

where

$$\begin{aligned} I_\gamma = \{t \in (0,T) : j({\bar{u}}(t)) = \gamma \} \ \ \text { and }\ \ I^+_\gamma = \{t \in I_\gamma : {\bar{\mu }}(t) \not \equiv 0 \text { in } \varOmega \}. \end{aligned}$$

We first prove the second order necessary conditions. Given an element \(v \in C_{{\bar{u}}}\), the classical approach to prove these second order conditions consists of taking a sequence \(\{v_k\}_{k = 1}^\infty \) converging to v such that \({\bar{u}} + \rho v_k\) is a feasible control for (P) for every \(\rho > 0\) small enough. The way of taking this sequence is different from the case where box control constraints are considered. The main reason for this difference is that the functional j, defining the constraint, is not differentiable and that it is non-local in space. Even the approach followed in the case where j is involved in the cost functional cannot be used in our framework; see [3]. The proof makes an essential use of the following lemma.

Lemma 5.1

Let \(v \in L^2(Q)\) satisfy \(j'({\bar{u}}(t);v(t)) = 0\) for almost all \(t \in I_\gamma ^+\). Then, \(J'({\bar{u}})v = 0\) holds if and only if

$$\begin{aligned} \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}|v(x,t)| = {\bar{\mu }}(x,t)v(x,t)\text { for a.a. } (x,t) \in \varOmega _{{\bar{u}}(t)}^0 \times I^+_\gamma . \end{aligned}$$
(5.2)

As a consequence, every element v of \(C_{{\bar{u}}}\) satisfies (5.2).

Proof

From (2.20), (3.5), and (3.8) we infer

$$\begin{aligned} J'({\bar{u}})v&= \int _Q({\bar{\varphi }} + \kappa {\bar{u}})v\,\mathrm {d}x\,\mathrm {d}t= -\int _Q{\bar{\mu }} v\,\mathrm {d}x\,\mathrm {d}t= -\int _{I_\gamma ^+}\int _\varOmega {\bar{\mu }} v\,\mathrm {d}x\,\mathrm {d}t\\&= -\int _{I^+_\gamma }\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\left\{ \int _{\varOmega _{{\bar{u}}(t)}^+}v \,\mathrm {d}x- \int _{\varOmega _{{\bar{u}}(t)}^-}v \,\mathrm {d}x\right\} - \int _{I^+_\gamma }\int _{\varOmega ^0_{{\bar{u}}(t)}}{\bar{\mu }} v\,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$

Using that \(j'({\bar{u}}(t);v(t)) = 0\) for almost all \(t \in I_\gamma ^+\) and (5.1) we get

$$\begin{aligned} \int _{\varOmega _{{\bar{u}}(t)}^+}v \,\mathrm {d}x- \int _{\varOmega _{{\bar{u}}(t)}^-}v \,\mathrm {d}x= -\int _{\varOmega ^0_{{\bar{u}}(t)}}|v|dx. \end{aligned}$$

Inserting this in the previous identity we obtain

$$\begin{aligned} J'({\bar{u}})v = \int _{I_\gamma ^+}\int _{\varOmega ^0_{{\bar{u}}(t)}}[\Vert \mu (t)\Vert _{L^\infty (\varOmega )}|v| - {\bar{\mu }} v]\,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$

Since \({\bar{\mu }} v \le \Vert \mu (t)\Vert _{L^\infty (\varOmega )}|v|\), we deduce from the above equality that \(J'({\bar{u}})v = 0\) if and only if (5.2) holds. \(\square \)

Theorem 5.1

Let \({\bar{u}}\) be a local solution of (P) in the \(L^r(0,T;L^2(\varOmega ))\) sense with \(r > \frac{4}{4-n}\). Then, the inequality \(J''({\bar{u}})v^2 \ge 0\) holds for all \(v \in C_{{\bar{u}}}\).

Proof

Let v be an element of \(C_{{\bar{u}}} \cap L^\infty (0,T;L^2(\varOmega ))\). We will prove that \(J''({\bar{u}})v^2 \ge 0\). Later, we will remove the assumption \(v \in L^\infty (0,T;L^2(\varOmega ))\). Set

$$\begin{aligned} g(x,t) = \left\{ \begin{array}{cl}\displaystyle \frac{v(x,t)}{|{\bar{u}}(x,t)|}&{}\text {if } x \not \in \varOmega _{{\bar{u}}(t)}^0,\\ 0 &{}\text {otherwise,}\end{array}\right. \quad \text { and }\quad a(t) = \int _\varOmega g(x,t){\bar{u}}(x,t)\,\mathrm {d}x. \end{aligned}$$

From (5.1) we infer

$$\begin{aligned} j'({\bar{u}}(t);v(t)) = a(t) + \int _{\varOmega _{{\bar{u}}(t)}^0}|v(x,t)|\,\mathrm {d}x. \end{aligned}$$

For every integer \(k \ge 1\) we put

$$\begin{aligned}&a_k(t) = \int _\varOmega {\text {Proj}}_{[-k,+k]}(g(x,t)){\bar{u}}(x,t)\,\mathrm {d}x,\\&g_k(x,t) = {\text {Proj}}_{[-k,+k]}(g(x,t))|{\bar{u}}(x,t)| + \frac{a(t) - a_k(t)}{\gamma }{\bar{u}}(x,t),\\&v_k(x,t) = \left\{ \begin{array}{cl}0&{}\text {if } \gamma - \frac{1}{k}< \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} < \gamma ,\\ g_k(x,t) + v(x,t)\chi _{\varOmega _{{\bar{u}}(t)}^0}(x)&{}\text {if } \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} = \gamma ,\\ v(x,t) &{} \text {otherwise,}\end{array}\right. \end{aligned}$$

where \(\chi _{\varOmega _{{\bar{u}}(t)}^0}(x)\) takes the value 1 if \(x \in \varOmega _{{\bar{u}}(t)}^0\) and 0 otherwise.

Using that \(|{\text {Proj}}_{[-k,+k]}(g(x,t)){\bar{u}}(x,t)| \le |v(x,t)|\) and the pointwise convergence \({\text {Proj}}_{[-k,+k]}(g(x,t)){\bar{u}}(x,t) \rightarrow g(x,t){\bar{u}}(x,t)\) almost everywhere in Q, we deduce with Lebesgue’s Theorem that \(\lim _{k \rightarrow \infty }a_k(t) = a(t)\) for almost all \(t \in (0,T)\). Therefore, we have that \(v_k(x,t) \rightarrow v(x,t)\) for almost all \((x,t) \in Q\). Moreover, we have

$$\begin{aligned} |g_k(x,t)| \le |v(x,t)| + \frac{2}{\gamma }\Vert v\Vert _{L^\infty (0,T;L^1(\varOmega ))}\Vert {\bar{u}}\Vert _{L^\infty (Q)} \end{aligned}$$

and

$$\begin{aligned} |v_k(x,t)| \le |v(x,t)| + \frac{2}{\gamma }\Vert v\Vert _{L^\infty (0,T;L^1(\varOmega ))}\Vert {\bar{u}}\Vert _{L^\infty (Q)} \text { for a.a. } (x,t) \in Q. \end{aligned}$$

Once again, with Lebesgue’s Theorem we get \(v_k \rightarrow v\) in \(L^r(0,T;L^2(\varOmega ))\) for every \(r < \infty \).

Let us prove that \(J'({\bar{u}})v_k = 0\). To this end, we apply Lemma 5.1. Actually, we are going to prove that \(v_k \in C_{{\bar{u}}}\). Given \(t \in I_\gamma \), taking into account (5.1) and the fact that \(j({\bar{u}}(t)) = \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} = \gamma \) we get

$$\begin{aligned}&j'({\bar{u}}(t);v_k(t))\\&= \int _{\varOmega ^+_{{\bar{u}}(t)}}{\text {Proj}}_{[-k,+k]}(g(x,t))|{\bar{u}}(x,t)|\,\mathrm {d}x- \int _{\varOmega ^-_{{\bar{u}}(t)}}{\text {Proj}}_{[-k,+k]}(g(x,t))|{\bar{u}}(x,t)|\,\mathrm {d}x\\&\quad + \frac{a(t)-a_k(t)}{\gamma }\Big [\int _{\varOmega ^+_{{\bar{u}}(t)}}{\bar{u}}(x,t)\,\mathrm {d}x- \int _{\varOmega ^-_{{\bar{u}}(t)}}{\bar{u}}(x,t)\,\mathrm {d}x\Big ] + \int _{\varOmega _{{\bar{u}}(t)}^0}|v(x,t)|\,\mathrm {d}x\\&= \int _\varOmega {\text {Proj}}_{[-k,+k]}(g(x,t)){\bar{u}}(x,t)\,\mathrm {d}x+ \frac{a(t)-a_k(t)}{\gamma }j({\bar{u}}(t)) + \int _{\varOmega _{{\bar{u}}(t)}^0}|v(x,t)|\,\mathrm {d}x\\&= a(t) + \int _{\varOmega _{{\bar{u}}(t)}^0}|v(x,t)|\,\mathrm {d}x= j'({\bar{u}}(t),v(t)) \left\{ \begin{array}{cl}= 0&{}\text {if } t \in I^+_\gamma ,\\ \le 0&{}\text {if } t \in I_\gamma \setminus I^+_\gamma ,\end{array}\right. \end{aligned}$$

where we used that \(v \in C_{{\bar{u}}}\) in the last step.

In the case where \(\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} < \gamma \), according to the definition of \(v_k\), we have that \(v_k(x,t)\) is equal to 0 or to v(xt). Since v satisfies (5.2) due to the fact that \(v \in C_{{\bar{u}}}\), we deduce that \(v_k\) also satisfies (5.2). Then, Lemma 5.1 implies that \(J'({\bar{u}})v_k = 0\). Therefore, \(v_k \in C_{{\bar{u}}}\) holds.

Take \(\rho _k > 0\) such that

$$\begin{aligned} \rho _k\Big (k + \frac{2}{\gamma }\Vert v\Vert _{L^\infty (0,T;L^1(\varOmega ))}\Big ) < \frac{1}{k\max \{1,\gamma \}}. \end{aligned}$$

Then, we have for each fixed k and \(\forall \rho \in (0,\rho _k)\)

$$\begin{aligned} \rho \Big (|{\text {Proj}}_{[-k,+k]}(g(x,t))| + \big |\frac{|a(t) - a_k(t)|}{\gamma }\big |\Big ) \le \rho \Big (k + \frac{2}{\gamma }\Vert v\Vert _{L^\infty (0,T;L^1(\varOmega ))}\Big ) < \frac{1}{k}. \end{aligned}$$

Using this estimate we have that \(\Vert {\bar{u}}(t) + \rho v_k(t)\Vert \le \gamma \) if \(j({\bar{u}}(t)) = \gamma \) and \(0< \rho < \rho _k\):

$$\begin{aligned}&\Vert {\bar{u}}(t) + \rho v_k(t)\Vert _{L^1(\varOmega )}\\&= \int _{\varOmega \setminus \varOmega ^0_{{\bar{u}}(t)}}\Big |{\bar{u}}(t)[1 + \rho \big [{\text {Proj}}_{[-k,+k]}(g(x,t)){\text {sign}}({\bar{u}}(x,t)) + \frac{a(t) - a_k(t)}{\gamma }\big ]\Big |\,\mathrm {d}x\\&\quad + \rho \int _{\varOmega ^0_{{\bar{u}}(t)}}|v(x,t)|\,\mathrm {d}x\\&= \int _{\varOmega \setminus \varOmega ^0_{{\bar{u}}(t)}}|{\bar{u}}(t)|[1 + \rho \big [{\text {Proj}}_{[-k,+k]}(g(x,t)){\text {sign}}({\bar{u}}(x,t)) + \frac{a(t) - a_k(t)}{\gamma }\big ]\,\mathrm {d}x\\&\quad + \rho \int _{\varOmega ^0_{{\bar{u}}(t)}}|v(x,t)|\,\mathrm {d}x\\&= \int _\varOmega |{\bar{u}}(t)|\,\mathrm {d}x+ \rho \left\{ \int _\varOmega \big [{\text {Proj}}_{[-k,+k]}(g(x,t)){\bar{u}}(x,t) + \frac{a(t) - a_k(t)}{\gamma }|{\bar{u}}(x,t)|\big ]\,\mathrm {d}x\right. \\&\quad \left. + \int _{\varOmega ^0_{{\bar{u}}(t)}}|v(x,t)|\,\mathrm {d}x\right\} \\&=\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} + \rho \left\{ a(t) + \int _{\varOmega ^0_{{\bar{u}}(t)}}|v(x,t)|\,\mathrm {d}x\right\} = \gamma + \rho j'({\bar{u}}(t);v(t)) \le \gamma . \end{aligned}$$

In the case \(\gamma - \frac{1}{k}< \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} < \gamma \), we have that \(v_k(t) = 0\) and, consequently

$$\begin{aligned} \Vert {\bar{u}}(t) + \rho v_k(t)\Vert _{L^1(\varOmega )} = \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} < \gamma . \end{aligned}$$

If \(\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} < \gamma - \frac{1}{k}\), then we get

$$\begin{aligned} \Vert {\bar{u}}(t) + \rho v_k(t)\Vert _{L^1(\varOmega )} \le \gamma - \frac{1}{k} + \rho \Vert v\Vert _{L^\infty (0,T;L^1(\varOmega ))} < \gamma . \end{aligned}$$

Using the local optimality of \({\bar{u}}\), the fact that \({\bar{u}} + \rho v_k \in {U_{ad}}\), \(J'({\bar{u}})v_k = 0\), and making a Taylor expansion we get for every \(\rho < \rho _k\) small enough

$$\begin{aligned} 0 \le J({\bar{u}} + \rho v_k) - J({\bar{u}}) = \rho J'({\bar{u}})v_k + \frac{\rho ^2}{2}J''({\bar{u}} + \theta \rho v_k)v_k^2 = \frac{\rho ^2}{2}J''({\bar{u}} + \theta \rho v_k)v_k^2. \end{aligned}$$

Dividing the above inequality by \(\rho ^2/2\) and making \(\rho \rightarrow 0\) we obtain with Corollary 2.1 that \(J''({\bar{u}})v_k^2 \ge 0\). Since \(v_k \rightarrow v\) in \(L^2(Q)\), we pass to the limit when \(k \rightarrow \infty \) and conclude that \(J''({\bar{u}})v^2 \ge 0\).

Finally, we take \(v \in C_{{\bar{u}}}\) arbitrary and for every \(k \ge 1\) set

$$\begin{aligned} v_k(x,t) = \frac{v(x,t)}{1 + \frac{1}{k}\Vert v(t)\Vert _{L^1(\varOmega )}}. \end{aligned}$$

Then, we have

$$\begin{aligned}&J'({\bar{u}})v_k = \frac{1}{1 + \frac{1}{k}\Vert v(t)\Vert _{L^1(\varOmega )}}J'({\bar{u}})v = 0\ \text { and }\\&j'({\bar{u}}(t);v_k(t)) = \frac{1}{1 + \frac{1}{k}\Vert v(t)\Vert _{L^1(\varOmega )}}j'({\bar{u}}(t);v(t)) \left\{ \begin{array}{cl} = 0&{}\text {if } t \in I_\gamma ^+,\\ \le 0&{}\text {if } t \in I_\gamma \setminus I^+_\gamma .\end{array}\right. \end{aligned}$$

Therefore, \(v_k \in C_{{\bar{u}}} \cap L^\infty (0,T;L^1(\varOmega ))\) and \(v_k \rightarrow v\) in \(L^2(Q)\) is satisfied. Hence, we get \(J''({\bar{u}})v^2 = \lim _{k \rightarrow \infty }J''({\bar{u}})v_k^2 \ge 0\), which concludes the proof. \(\square \)

Theorem 5.2

Let \({\bar{u}} \in {U_{ad}}\cap L^\infty (Q)\) satisfy the first order optimality conditions (3.2)–(3.5). If \(J''({\bar{u}})v^2 > 0\) \(\forall v \in C_{{\bar{u}}} \setminus \{0\}\) holds, then for each \(r \in \big (\frac{4}{4 - n},\infty ]\) there exist \(\delta > 0\) and \(\varepsilon > 0\) such that

$$\begin{aligned} J({\bar{u}}) + \frac{\delta }{2}\Vert u - {\bar{u}}\Vert ^2_{L^2(Q)} \le J(u)\quad \forall u \in {U_{ad}}\cap B_\varepsilon ({\bar{u}}), \end{aligned}$$
(5.3)

where \(B_\varepsilon ({\bar{u}}) = \{u \in L^r(0,T;L^2(\varOmega )) : \Vert u - {\bar{u}}\Vert _{L^r(0,T;L^2(\varOmega ))} \le \varepsilon \}\).

Proof

We proceed by contradiction. If (5.3) is false for every \(\delta > 0\) and \(\varepsilon > 0\), then for every integer \(k \ge 1\) there exists an element \(u_k \in {U_{ad}}\) such that

$$\begin{aligned} \Vert u_k - {\bar{u}}\Vert _{L^r(0,T;L^2(\varOmega ))}< \frac{1}{k}\ \text { and } \ J(u_k) < J({\bar{u}}) + \frac{1}{2k}\Vert u_k - {\bar{u}}\Vert ^2_{L^2(Q)}. \end{aligned}$$
(5.4)

Let us set \(\rho _k = \Vert u_k - {\bar{u}}\Vert _{L^2(Q)}\) and \(v_k = (u_k - {\bar{u}})/\rho _k\). Then, we have \(\Vert v_k\Vert _{L^2(Q)} = 1\) and, taking a subsequence that we denote in the same way, we have \(v_k \rightharpoonup v\) in \(L^2(Q)\). We divide the proof in several steps.

Step I - \(J'({\bar{u}})v = 0\). From (3.4) and (3.5) we infer that \(J'({\bar{u}})(u_k - {\bar{u}}) \ge 0\) for every \(k \ge 1\). Therefore, \(J'({\bar{u}})v_k \ge 0\) and passing to the limit we obtain \(J'({\bar{u}})v \ge 0\). Now, using (5.4) along with the mean value theorem we get for some \(\theta _k \in (0,1)\)

$$\begin{aligned} J(u_k) - J({\bar{u}}) = J'({\bar{u}} + \theta _k(u_k - {\bar{u}}))(u_k - {\bar{u}}) < \frac{1}{2k}\Vert u_k - {\bar{u}}\Vert ^2_{L^2(Q)}. \end{aligned}$$

Dividing this inequality by \(\rho _k\) we obtain

$$\begin{aligned} J'({\bar{u}} + \theta _k(u_k - {\bar{u}}))v_k < \frac{1}{2k}\Vert u_k - {\bar{u}}\Vert _{L^2(Q)}. \end{aligned}$$

Then, passing to the limit when \(k \rightarrow \infty \) it follows \(J'({\bar{u}})v \le 0\).

Step II - \(v \in C_{{\bar{u}}}\). Since \({\bar{u}}(t) + \lambda v_k(t) = {\bar{u}}(t) + \frac{\lambda }{\rho _k}(u_k(t) - {\bar{u}}(t)) \in {U_{ad}}\) for every \(0< \lambda < \rho _k\), we get for almost every \(t \in I_\gamma \)

$$\begin{aligned} j'({\bar{u}}(t);v_k(t))&= \lim _{\lambda \searrow 0}\frac{\Vert {\bar{u}}(t) + \lambda v_k(t)\Vert _{L^1(\varOmega )} - \Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )}}{\lambda }\\&= \lim _{\lambda \searrow 0}\frac{\Vert {\bar{u}}(t) + \lambda v_k(t)\Vert _{L^1(\varOmega )} - \gamma }{\lambda } \le 0. \end{aligned}$$

Take a measurable subset \(J \subset I_\gamma \). Since the functional

$$\begin{aligned} u \in L^2(Q) \longrightarrow \int _Jj'({\bar{u}}(t);u(t))\,\mathrm {d}t\in {\mathbb {R}} \end{aligned}$$

is continuous and convex, recall (5.1), the weak convergence \(v_k \rightharpoonup v\) in \(L^2(Q)\) implies

$$\begin{aligned} \int _Jj'({\bar{u}}(t);v(t))\,\mathrm {d}t\le \liminf _{k \rightarrow \infty }\int _Jj'({\bar{u}}(t);v_k(t))\,\mathrm {d}t\le 0. \end{aligned}$$

Since \(J \subset I_\gamma \) is an arbitrary measurable set, we infer for almost all \(t \in I_\gamma \)

$$\begin{aligned} \int _{\varOmega ^+_{{\bar{u}}(t)}}v(t) \,\mathrm {d}x- \int _{\varOmega ^-_{{\bar{u}}(t)}}v(t) \,\mathrm {d}x+\int _{\varOmega ^0_{{\bar{u}}(t)}}|v(t)| \,\mathrm {d}x= j'({\bar{u}}(t);v(t)) \le 0. \end{aligned}$$
(5.5)

Identities (3.5) and \(J'({\bar{u}})v = 0\), and (3.8) imply

$$\begin{aligned}&0 = \int _Q{\bar{\mu }}(x,t) v(x,t)\,\mathrm {d}x\,\mathrm {d}t= \int _{I^+_\gamma }\int _\varOmega {\bar{\mu }}(x,t) v(x,t)\,\mathrm {d}x\,\mathrm {d}t\nonumber \\&=\int _{I^+_\gamma }\left\{ \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\Big [\int _{\varOmega ^+_{{\bar{u}}(t)}}v(t) \,\mathrm {d}x- \int _{\varOmega ^-_{{\bar{u}}(t)}}v(t) \,\mathrm {d}x\Big ] + \int _{\varOmega ^0_{{\bar{u}}(t)}}{\bar{\mu }}(t)v(t) \,\mathrm {d}x\right\} \,\mathrm {d}t. \end{aligned}$$
(5.6)

From (5.5) we deduce

$$\begin{aligned} \int _{I^+_\gamma }\left\{ \Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\Big [\int _{\varOmega ^+_{{\bar{u}}(t)}}v(t) \,\mathrm {d}x- \int _{\varOmega ^-_{{\bar{u}}(t)}}v(t) \,\mathrm {d}x+\int _{\varOmega ^0_{{\bar{u}}(t)}}|v(t)| \,\mathrm {d}x\Big ]\right\} \,\mathrm {d}t\le 0. \end{aligned}$$

The last two relations lead to

$$\begin{aligned} \int _{I^+_\gamma }\left\{ \int _{\varOmega ^0_{{\bar{u}}(t)}}\Big [\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}|v(t)| - {\bar{\mu }}(t)v(t)\Big ]\,\mathrm {d}x\right\} \,\mathrm {d}t\le 0. \end{aligned}$$

This is possible if and only if \(\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}|v(x,t)| = {\bar{\mu }}(x,t)v(x,t)\) for almost all \(t \in I^+_\gamma \) and \(x \in \varOmega ^0_{{\bar{u}}(t)}\). Inserting this identity in (5.6) we get

$$\begin{aligned} 0 = \int _Q{\bar{\mu }}(x,t) v(x,t)\,\mathrm {d}x\,\mathrm {d}t= \int _{I^+_\gamma }\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}j'({\bar{u}}(t);v(t))\,\mathrm {d}t. \end{aligned}$$

Finally, this identity and (5.5) yield \(j'({\bar{u}}(t);v(t)) = 0\) for almost all \(t \in I^+_\gamma \). Therefore, we conclude with Step I that \(v \in C_{{\bar{u}}}\).

Step III - \(J''({\bar{u}})v^2 \le 0\). From (5.4) and a Taylor expansion we infer

$$\begin{aligned} \rho _kJ'({\bar{u}})v_k + \frac{\rho _k^2}{2}J''({\bar{u}} + \theta _k\rho _kv_k)v_k^2 = J(u_k) - J({\bar{u}}) < \frac{1}{2k}\Vert u_k - {\bar{u}}\Vert ^2_{L^2(Q)}. \end{aligned}$$

Since \(J'({\bar{u}})v_k = \frac{1}{\rho _k}J'({\bar{u}})(u_k - {\bar{u}}) \ge 0\), we deduce from the above inequality

$$\begin{aligned} J''({\bar{u}} + \theta _k(u_k - {\bar{u}}))v_k^2 = J''({\bar{u}} + \theta _k\rho _kv_k)v_k^2 < \frac{1}{k}. \end{aligned}$$
(5.7)

The strong convergence \({\bar{u}} + \theta _k(u_k - {\bar{u}}) \rightarrow {\bar{u}}\) in \(L^r(0,T;L^2(\varOmega ))\) yields the uniform convergences \(y_{\theta _k} \rightarrow {\bar{y}}\) and \(\varphi _{\theta _k} \rightarrow {\bar{\varphi }}\) in \(L^\infty (Q)\), where \(y_{\theta _k}\) and \(\varphi _{\theta _k}\) are the state and adjoint state associated with \({\bar{u}} + \theta _k(u_k - {\bar{u}})\). This also implies that \(z_{\theta _k,v_k} \rightarrow z_v\) strongly in \(L^2(Q)\), where \(z_v\) is the solution of (2.20) for \(y_u = {\bar{y}}\) and \(z^2_{\theta _k,v_k}\) is the solution of (2.20) with \(v = v_k\) and \(y_u = y_{\theta _k}\). Then, we can pass to the limit in (5.7) when \(k \rightarrow \infty \) and deduce that \(J''({\bar{u}})v^2 \le 0\).

Step IV - Final contradiction. Since \(v \in C_{{\bar{u}}}\) and \(J''({\bar{u}})v^2 \le 0\), according to the assumptions of the theorem, this is only possible if \(v = 0\). Therefore, we have that \(v_k \rightharpoonup 0\) in \(L^2(Q)\) and, consequently, \(z^2_{\theta _k,v_k} \rightarrow 0\) strongly in \(L^2(Q)\). Now, using that \(\Vert v_k\Vert _{L^2(Q)} = 1\) and (2.21), we infer from (5.7)

$$\begin{aligned}&0 \ge \liminf _{k \rightarrow \infty }J''({\bar{u}} + \theta _k(u_k - {\bar{u}}))v_k^2\\&= \liminf _{k \rightarrow \infty }\int _Q\Big [\big (1 - \frac{\partial ^2a}{\partial y^2}(x,t,y_{\theta _k})\varphi _{\theta _k}\big )z^2_{\theta _k,v_k} + \kappa v_k^2\Big ]\,\mathrm {d}x\,\mathrm {d}t\\&\lim _{k \rightarrow \infty }\int _Q\big (1 - \frac{\partial ^2a}{\partial y^2}(x,t,y_{\theta _k})\varphi _{\theta _k}\big )z^2_{\theta _k,v_k}\,\mathrm {d}x\,\mathrm {d}t+ \kappa = \kappa , \end{aligned}$$

which contradicts our assumption \(\kappa > 0\). \(\square \)

The next theorem establishes that the sufficient condition for local optimality, \(J''({\bar{u}})v^2 > 0\) for every \(v \in C_{{\bar{u}}} \setminus \{0\}\), provides a useful tool for the numerical analysis of the control problem. Given \(\tau > 0\) we define the extended cone

$$\begin{aligned}&C^\tau _{{\bar{u}}} = \Bigg \{v \in L^2(Q) : |J'({\bar{u}})v| \le \tau \Vert v\Vert _{L^2(Q)} \text { and }\\&\Bigg \{\begin{array}{cl}|j'({\bar{u}}(t);v(t))| \le \tau \Vert v\Vert _{L^2(Q)}&{}\text {if } t \in I^+_\gamma ,\\ j'({\bar{u}}(t);v(t)) \le \tau \Vert v\Vert _{L^2(Q)}&{}\text {if } t \in I_\gamma \setminus I^+_\gamma ,\end{array}\ \ \Bigg \}. \end{aligned}$$

Theorem 5.3

Let \({\bar{u}} \in {U_{ad}}\) satisfy the first order optimality conditions (3.2)–(3.5) and the second order condition \(J''({\bar{u}})v^2 > 0\) \(\forall v \in C_{{\bar{u}}} \setminus \{0\}\). Then, for every \(r \in \big (\frac{4}{4-n},\infty ]\) there exist strictly positive numbers \(\varepsilon , \tau , \nu \) such that

$$\begin{aligned} J''(u)v^2 \ge \nu \Vert v\Vert ^2_{L^2(Q)}\quad \forall v \in C^\tau _{{\bar{u}}}\ \text { and } \ \forall u \in B_\varepsilon ({\bar{u}}), \end{aligned}$$
(5.8)

where \(B_\varepsilon ({\bar{u}})\) denotes the \(L^r(0,T;L^2(\varOmega ))\) closed ball.

Proof

First we prove the existence of \(\tau > 0\) and \(\nu > 0\) such that

$$\begin{aligned} J''({\bar{u}})v^2 \ge 2\nu \Vert v\Vert ^2_{L^2(Q)} \quad \forall v \in C^\tau _{{\bar{u}}}. \end{aligned}$$
(5.9)

We proceed by contradiction. If (5.9) fails for all strictly positive numbers \(\tau , \nu \), then for every integer \(k \ge 1\) there exists a function \(v_k \in C^{\frac{1}{k}}_{{\bar{u}}}\) such that \(J''({\bar{u}})v_k^2 < \frac{1}{k}\Vert v_k\Vert ^2_{L^2(Q)}\). Dividing \(v_k\) by its \(L^2(Q)\) norm and taking a subsequence we get

$$\begin{aligned}&\Vert v_k\Vert _{L^2(Q)} = 1,\quad v_k \rightharpoonup v \text { in } L^2(Q),\quad J''({\bar{u}})v_k^2 < \frac{1}{k}, \end{aligned}$$
(5.10)
$$\begin{aligned}&|J'({\bar{u}})v_k| \le \frac{1}{k},\quad \left\{ \begin{array}{cl}|j'({\bar{u}}(t);v_k(t))| \le \frac{1}{k}&{}\text {if } t \in I^+_\gamma ,\\ j'({\bar{u}}(t);v_k(t)) \le \frac{1}{k}&{}\text {if } t \in I_\gamma \setminus I^+_\gamma .\end{array}\right. \end{aligned}$$
(5.11)

We prove that \(v \in C_{{\bar{u}}}\). First, from (5.10) and (5.11) we get

$$\begin{aligned} |J'({\bar{u}})v| \le \liminf _{k \rightarrow \infty }|J'({\bar{u}})v_k| \le 0. \end{aligned}$$

Thus, we have \(J'({\bar{u}})v = 0\). Let us set

$$\begin{aligned} I = \{t \in I_\gamma : j'({\bar{u}}(t);v(t)) > 0\}. \end{aligned}$$

Then, we obtain with (5.10) and (5.11)

$$\begin{aligned} \int _Ij'({\bar{u}}(t);v(t))\,\mathrm {d}t\le \liminf _{k \rightarrow \infty }\int _Ij'({\bar{u}}(t);v_k(t))\,\mathrm {d}t\le 0. \end{aligned}$$

This is not possible unless \(|I| = 0\). Hence, we have that \(j'({\bar{u}}(t);v(t)) \le 0\) for almost all \(t \in I_\gamma \). Now, from the identity \(J'({\bar{u}})v = 0\), (5.1), and (3.8) it follows

$$\begin{aligned}&0 = \int _Q({\bar{\varphi }} + \kappa {\bar{u}})v\,\mathrm {d}x\,\mathrm {d}t= -\int _Q{\bar{\mu }} v\,\mathrm {d}x\,\mathrm {d}t\\&= -\int _{I^+_\gamma }\left[ \int _{\varOmega ^+_{{\bar{u}}(t)}}\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}v\,\mathrm {d}x- \int _{\varOmega ^-_{{\bar{u}}(t)}}\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}v\,\mathrm {d}x+ \int _{\varOmega ^0_{{\bar{u}}(t)}}\mu v\,\mathrm {d}x\right] \,\mathrm {d}t. \end{aligned}$$

This implies

$$\begin{aligned}&\int _{I^+_\gamma }\left[ \int _{\varOmega ^+_{{\bar{u}}(t)}}\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}v\,\mathrm {d}x- \int _{\varOmega ^-_{{\bar{u}}(t)}}\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}v\,\mathrm {d}x\right] \,\mathrm {d}t\nonumber \\&\quad = - \int _{I^+_\gamma }\int _{\varOmega ^0_{{\bar{u}}(t)}}\mu v \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
(5.12)

Now we have

$$\begin{aligned}&\int _{I^+_\gamma }\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )} j'({\bar{u}}(t);v(t))\,\mathrm {d}t\\&=\int _{I^+_\gamma }\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}\left[ \int _{\varOmega ^+_{{\bar{u}}(t)}}v\,\mathrm {d}x- \int _{\varOmega ^-_{{\bar{u}}(t)}}v\,\mathrm {d}x+ \int _{\varOmega ^0_{{\bar{u}}(t)}}|v| \,\mathrm {d}x\right] \,\mathrm {d}t. \end{aligned}$$

From this identity and (5.12) we infer

$$\begin{aligned}&\int _{I^+_\gamma }\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )} j'({\bar{u}}(t);v(t))\,\mathrm {d}t= \int _{I^+_\gamma }\int _{\varOmega ^0_{{\bar{u}}(t)}}\big [\Vert {\bar{\mu }}(t)\Vert _{L^\infty (\varOmega )}|v| - {\bar{\mu }}(t)v\big ]\,\mathrm {d}x\,\mathrm {d}t\ge 0. \end{aligned}$$

This inequality along with \(j'({\bar{u}}(t);v(t)) \le 0\) for \(t \in I_\gamma \) implies that \(j'({\bar{u}}(t);v(t)) = 0\) for almost all \(t \in I^+_\gamma \). We have proved that \(v \in C_{{\bar{u}}}\). From (5.10) we infer

$$\begin{aligned} J''({\bar{u}})v^2 \le \liminf _{k \rightarrow \infty }J''({\bar{u}})v_k^2 \le 0. \end{aligned}$$

Since \({\bar{u}}\) satisfies the second order condition, the above inequality is only possible if \(v = 0\). Therefore, we have that \(v_k \rightharpoonup 0\) in \(L^2(Q)\). Using (2.21) and the fact that \(z_{v_k}\rightarrow 0\) strongly in \(L^2(Q)\) this yields

$$\begin{aligned} \kappa = \liminf _{k \rightarrow \infty } \kappa \Vert v_k\Vert ^2_{L^2(Q)} = \liminf _{k \rightarrow \infty }J''({\bar{u}})v_k^2 \le 0, \end{aligned}$$

which is a contradiction. Therefore, (5.9) holds.

Let us conclude the proof showing that (5.9) implies (5.8). Given \(\rho > 0\) arbitrarily small, from Theorem 2.2 we deduce the existence of \(\varepsilon > 0\) such that

$$\begin{aligned} \Vert y_u - {\bar{y}}\Vert _{L^\infty (Q)} = \Vert G(u) - G({\bar{u}})\Vert _{L^\infty (Q)} < \rho \quad \forall u \in B_\varepsilon ({\bar{u}}). \end{aligned}$$

Using this estimate, we get from (2.17) and (2.22), and taking a smaller \(\varepsilon \) if necessary

$$\begin{aligned} \Vert \varphi _u - {\bar{\varphi }}\Vert _{L^\infty (Q)} + \Vert z_{u,v} - z_v\Vert _{L^2(Q)} < \rho \quad \forall u \in B_\varepsilon ({\bar{u}})\ \text { and }\ \forall v \in L^2(Q), \end{aligned}$$

where \(z_{u,v} = G'(u)v\), \(z_v = G'({\bar{u}})v\), and \(\varphi _u\) and \({\bar{\varphi }}\) are the adjoint states corresponding to u and \({\bar{u}}\), respectively. Therefore, selecting \(\rho \) small enough we obtain with (2.21) for some \(\varepsilon > 0\)

$$\begin{aligned} |[J''(u) - J''({\bar{u}})]v^2| \le \nu \Vert v\Vert ^2_{L^2(Q)} \quad \forall u \in B_\varepsilon ({\bar{u}}) \ \text { and }\ \forall v \in L^2(Q). \end{aligned}$$

Combining this with (5.9) we infer (5.8). \(\square \)

6 Stability of the Optimal Controls with Respect to \(\gamma \)

The aim of this section is to prove some stability of the local or global solutions of (P) with respect to \(\gamma \). For every \(\gamma > 0\) we consider the control problems

$$\begin{aligned} \text{(P }_\gamma ) \quad \inf _{u \in {U_\gamma }\cap L^\infty (Q)} J(u), \end{aligned}$$

where

$$\begin{aligned} {U_\gamma }= \{u \in L^\infty (0,T;L^1(\varOmega )) : \Vert u(t)\Vert _{L^1(\varOmega )} \le \gamma \text { for a.a. } t \in (0,T)\}. \end{aligned}$$

First, we prove some continuity of the solutions of \( \text{(P }_\gamma ) \) with respect to \(\gamma \).

Theorem 6.1

Let \(\{\gamma _k\}_{k = 1}^\infty \subset (0,\infty )\) be a sequence converging to some \(\gamma > 0\). For every k let \(u_{\gamma _k}\) be a global minimizer of the problem \( \text{(P }_{\gamma _k}) \). Then, the sequence \(\{u_{\gamma _k}\}_{k = 1}^\infty \) is bounded in \(L^\infty (Q)\). Moreover, if \(u_\gamma \) is a \(\hbox {weak}^*\) limit in \(L^\infty (Q)\) of a subsequence of \(\{u_{\gamma _k}\}_{k = 1}^\infty \), then \(u_\gamma \) is a global minimizer of \( \text{(P }_\gamma ) \) and the convergence is strong in \(L^p(Q)\) for every \(p < \infty \). Reciprocally, for every strict local minimizer \(u_\gamma \) of \( \text{(P }_\gamma ) \) in the \(L^r(0,T;L^2(\varOmega ))\) sense with \(\frac{4}{4-n}< r < \infty \), there exists a sequence \(\{u_{\gamma _k}\}_{k = 1}^\infty \) such that \(u_{\gamma _k}\) is a \(L^r(0,T;L^2(\varOmega ))\) local minimizer of \( \text{(P }_{\gamma _k}) \) and \(u_{\gamma _k} \rightarrow u_\gamma \) strongly in \(L^p(Q)\) for every \(p < \infty \).

Proof

The boundedness of \(\{u_{\gamma _k}\}_{k = 1}^\infty \) in \(L^\infty (Q)\) follows from Theorem 3.2. Therefore, we can take subsequences converging \(\hbox {weakly}^*\) in \(L^\infty (Q)\). Let us take one of these subsequences, that we denote in the same form, such that \(u_{\gamma _k} {\mathop {\rightharpoonup }\limits ^{*}} {\hat{u}}\) in \(L^\infty (Q)\). Let \(u_\gamma \) be a solution of \( \text{(P }_\gamma ) \). For every k we define

$$\begin{aligned} u_k = \left\{ \begin{array}{cl} u_\gamma &{}\text {if } \gamma \le \gamma _k,\\ \frac{\gamma _k}{\gamma }u_\gamma &{}\text {if } \gamma> \gamma _k,\end{array}\right. \quad \text { and }\quad {\hat{u}}_k = \left\{ \begin{array}{cl} u_{\gamma _k}&{}\text {if } \gamma _k \le \gamma ,\\ \frac{\gamma }{\gamma _k}u_{\gamma _k}&{}\text {if } \gamma _k > \gamma .\end{array}\right. \end{aligned}$$
(6.1)

Then, it is immediate that \(u_k \rightarrow u_\gamma \) and \({\hat{u}}_k {\mathop {\rightharpoonup }\limits ^{*}} {\hat{u}}\) in \(L^\infty (Q)\), \(\{{\hat{u}}_k\}_{k = 1}^\infty \subset {U_\gamma }\) and \(u_k \in {U_{\gamma _k}}\cap {U_\gamma }\) for every k. Since \({U_\gamma }\cap L^2(Q)\) is a closed and convex subset of \(L^2(Q)\) and \({\hat{u}}_k \rightharpoonup {\hat{u}}\) in \(L^2(Q)\), we deduce that \({\hat{u}} \in {U_\gamma }\). With the compactness of the embedding \(W(0,T) \subset L^2(Q)\) we can easily prove that \(y_{{\hat{u}}_k} \rightarrow y_{{\hat{u}}}\) in \(L^2(Q)\). Using these convergences and the optimality of \(u_{\gamma _k}\) and \(u_\gamma \) we get

$$\begin{aligned} J(u_\gamma ) \le J({\hat{u}}) \le \liminf _{k \rightarrow \infty }J(u_{\gamma _k}) \le \limsup _{k \rightarrow \infty }J(u_{\gamma _k}) \le \limsup _{k \rightarrow \infty } J(u_k) = J(u_\gamma ). \end{aligned}$$

This implies that \(J(u_\gamma ) = J({\hat{u}}) = \lim _{k \rightarrow \infty }J(u_{\gamma _k})\). This identity proves that \({\hat{u}}\) is a solution of \( \text{(P }_\gamma ) \). Moreover, the convergence \(y_{u_{\gamma _k}} \rightarrow y_{u_\gamma }\) in \(L^2(Q)\) leads to \(\lim _{k \rightarrow \infty }\Vert u_{\gamma _k}\Vert _{L^2(Q)} = \Vert {\hat{u}}\Vert _{L^2(Q)}\). From this fact and the weak convergence \(u_{\gamma _k} \rightharpoonup u_\gamma \) in \(L^2(Q)\), we obtain that \(u_{\gamma _k} \rightarrow {\hat{u}}\) in \(L^2(Q)\). This along with the boundedness of \(\{u_{\gamma _k}\}_{k = 1}^\infty \) in \(L^\infty (Q)\) implies the strong convergence in \(L^p(Q)\) for every \(p < \infty \).

Let us prove the second part of the theorem. Let \(u_\gamma \) be an \(L^r(0,T;L^2(\varOmega ))\) strict local minimizer to \( \text{(P }_\gamma ) \). This means that there exists \(\varepsilon > 0\) such that

$$\begin{aligned} J(u_\gamma ) < J(u) \quad \forall u \in {U_\gamma }\cap B_\varepsilon (u_\gamma ) \text { with } u \ne u_\gamma , \end{aligned}$$

where \(B_\varepsilon (u_\gamma )\) is the closed ball in \(L^r(0,T;L^2(\varOmega ))\) of radius \(\varepsilon \) and center \(u_\gamma \). Now, we consider the problems

$$\begin{aligned} (PB_\gamma ) \ \ \min _{u \in {U_\gamma }\cap B_\varepsilon (u_\gamma )}J(u) \qquad \text { and }\qquad (PB_{\gamma _k})\ \ \min _{u \in {U_{\gamma _k}}\cap B_\varepsilon (u_\gamma )}J(u) \end{aligned}$$

It is immediate that \(u_\gamma \) is the unique solution of \((PB_\gamma )\). Observe that the controls \(u_k\) defined in (6.1) are elements of \({U_{\gamma _k}}\cap B_\varepsilon (u_\gamma )\) for all k large enough. Hence, \({U_{\gamma _k}}\cap B_\varepsilon (u_\gamma )\) is non-empty, closed, convex, and bounded in \(L^r(0,T;L^2(\varOmega ))\). Therefore, problem \((PB_{\gamma _k})\) has at least one solution \(u_{\gamma _k}\). Let us prove that \(u_{\gamma _k} \rightarrow u_\gamma \) in \(L^p(Q)\) for every \(p < \infty \). Denote \(y_{\gamma _k}\) and \(\varphi _{\gamma _k}\) the state and adjoint state associated with \(u_{\gamma _k}\). Since \(\{u_{\gamma _k}\}_{k = 1}^\infty \) is bounded in \(L^r(0,T;L^2(\varOmega ))\) we infer from Theorem 2.1 the boundedness of \(\{y_{\gamma _k}\}_{k = 1}^\infty \) in \(L^\infty (Q)\). Hence, from the adjoint state equation and the classical estimates for linear equations we deduce that \(\{\varphi _{\gamma _k}\}_{k = 1}^\infty \) is also bounded in \(L^\infty (Q)\). Due to the optimality of \(u_{\gamma _k}\) for \((PB_{\gamma _k})\) we obtain

$$\begin{aligned} \int _Q(\varphi _{\gamma _k} + \kappa u_{\gamma _k})(u - u_{\gamma _k})\,\mathrm {d}x\,\mathrm {d}t= J'(u_{\gamma _k})(u - u_{\gamma _k}) \ge 0 \ \ \forall u \in {U_{\gamma _k}}\cap B_\varepsilon (u_\gamma ). \end{aligned}$$

Setting \(S = {U_{\gamma _k}}\cap B_\varepsilon (u_\gamma )\) we get from the above inequalities

$$\begin{aligned} u_{\gamma _k} = {\text {Proj}}_S\Big (-\frac{1}{\kappa }\varphi _{\gamma _k}\Big ), \end{aligned}$$

where \({\text {Proj}}_S\) denotes the \(L^2(Q)\) projection on S. Let us prove that

$$\begin{aligned} \Vert u_{\gamma _k}\Vert _{L^\infty (Q)} \le 2\Big (\frac{1}{\kappa }\Vert \varphi _{\gamma _k}\Vert _{L^\infty (Q)} + \Vert u_\gamma \Vert _{L^\infty (Q)}\Big ). \end{aligned}$$
(6.2)

For this purpose we define

$$\begin{aligned} Q_0 = \Big \{(x,t) \in Q : |u_{\gamma _k}(x,t)| > 2\Big (\frac{1}{\kappa }|\varphi _{\gamma _k}(x,t)| + |u_\gamma (x,t)|\Big )\Big \}. \end{aligned}$$

Put

$$\begin{aligned} u(x,t) = \left\{ \begin{array}{cl}-\frac{1}{\kappa }\varphi _{\gamma _k}(x,t) + u_\gamma (x,t) &{}\text {if } (x,t) \in Q_0,\\ u_{\gamma _k}(x,t)&{}\text {otherwise.}\end{array}\right. \end{aligned}$$

Then, it is obvious that

$$\begin{aligned}&\Vert u(t)\Vert _{L^1(\varOmega )} \le \Vert u_{\gamma _k}(t)\Vert _{L^1(\varOmega )} \le \gamma _k,\\&\Vert u - u_\gamma \Vert _{L^r(0,T;L^2(\varOmega ))} \le \Vert u_{\gamma _k} - u_\gamma \Vert _{L^r(0,T;L^2(\varOmega ))} \le \varepsilon ,\\&\big \Vert u + \frac{1}{\kappa }\varphi _{\gamma _k}\big \Vert _{L^2(Q)} < \big \Vert u_{\gamma _k} + \frac{1}{\kappa }\varphi _{\gamma _k}\big \Vert _{L^2(Q)}\ \text { if } |Q_0| \ne 0, \end{aligned}$$

The first two inequalities show that \(u \in S\) and, consequently, the third one contradicts the fact that \(u_{\gamma _k}\) is the \(L^2(Q)\) projection of \(-\frac{1}{\kappa }\varphi _{\gamma _k}\) unless \(|Q_0| = 0\). Now, the boundedness of \(\{\varphi _{\gamma _k}\}_{k = 1}^\infty \) in \(L^\infty (Q)\) and (6.2) imply the boundedness of \(\{u_{\gamma _k}\}_{k = 1}^\infty \). Therefore, there exists a subsequence, denoted in the same way, such that \(u_{\gamma _k} {\mathop {\rightharpoonup }\limits ^{*}} {\hat{u}}\) in \(L^\infty (Q)\). Using the functions \(\{{\hat{u}}_k\}_{k = 1}^\infty \) defined in (6.1) and arguing as above, we deduce that \({\hat{u}} \in {U_\gamma }\). Moreover, is is also immediate that \({\hat{u}} \in B_\varepsilon (u_\gamma )\). Let us consider the functions \(\{u_k\}_{k = 1}^\infty \) defined in (6.1). Since

$$\begin{aligned} \Vert u_k - u_\gamma \Vert _{L^\infty (Q)} = \left\{ \begin{array}{cl}0 &{}\text {if } \gamma \le \gamma _k,\\ \frac{\gamma - \gamma _k}{\gamma }\Vert u_\gamma \Vert _{L^\infty (Q)}&{}\text {otherwise,}\end{array}\right. \end{aligned}$$

we have that \(u_k \rightarrow u_\gamma \) in \(L^\infty (Q)\) as \(k \rightarrow \infty \) and \(u_k \in {U_{\gamma _k}}\cap B_\varepsilon (u_\gamma )\) for every k large enough. Then, using the optimality of \(u_\gamma \) and \(u_{\gamma _k}\), and the fact that \(u_k\) and \({\hat{u}}\) are feasible controls for \((PB_{\gamma _k})\) and \((PB_\gamma )\), respectively, we infer

$$\begin{aligned} J(u_\gamma ) \le J({\hat{u}}) \le \liminf _{k \rightarrow \infty }J(u_{\gamma _k}) \le \limsup _{k \rightarrow \infty }J(u_{\gamma _k}) \le \limsup _{k \rightarrow \infty } J(u_k) = J(u_\gamma ). \end{aligned}$$

This implies that \(J(u_\gamma ) = J({\hat{u}})\) and, hence, \({\hat{u}}\) is also a solution of \((PB_\gamma )\). Due to the uniqueness of solution of \((PB_\gamma )\) we conclude that \(u_\gamma = {\hat{u}}\). The strong convergence \(u_{\gamma _k} \rightarrow u_\gamma \) in \(L^p(Q)\) follows as above. We have proved that every subsequence converge to \(u_\gamma \), then the whole sequence does. In particular, the convergence \(u_{\gamma _k} \rightarrow u_\gamma \) in \(L^r(0,T;L^2(\varOmega ))\) implies that \(u_{\gamma _k}\) is in the interior of the ball \(B_\varepsilon (u_\gamma )\) for all k sufficiently large. Hence, \(u_{\gamma _k}\) is an \(L^r(0,T;L^2(\varOmega ))\) local minimizer of \((PB_{\gamma _k})\). \(\square \)

Remark 6.1

Given an \(L^r(0,T;L^2(\varOmega ))\) strict local minimizer of \( \text{(P }_\gamma ) \), from the above theorem we deduce the existence of a family \(\{u_{\gamma '}\}_{\gamma ' > 0}\) of \(L^r(0,T;L^2(\varOmega ))\) local minimizers of problems \( \text{(P }_{\gamma '}) \) such that \(u_{\gamma '} \rightarrow u_\gamma \) in \(L^p(Q)\) as \(\gamma ' \rightarrow \gamma \) for every \(p < \infty \). Looking at the definition of the elements \(u_{\gamma _k}\) in the previous proof we have that

$$\begin{aligned} J(u_{\gamma '}) \le J(u) \ \ \forall u \in {U_{\gamma '}}\cap B_\varepsilon (u_\gamma )\quad \text {and}\quad J(u_\gamma ) \le J(u) \ \ \forall u \in {U_\gamma }\cap B_\varepsilon (u_\gamma ).\nonumber \\ \end{aligned}$$
(6.3)

Theorem 6.2

Let \(\{u_{\gamma '}\}_{\gamma '}\) be a family of local minimizers of problems \( \text{(P }_{\gamma '}) \) such that \(u_{\gamma '} \rightarrow u_\gamma \) in \(L^r(0,T;L^2(\varOmega ))\) as \(\gamma ' \rightarrow \gamma \) with \(u_\gamma \) a local minimizer of \( \text{(P }_\gamma ) \) satisfying (5.3). We also assume that (6.3) holds. Then, there exists a constant L such that

$$\begin{aligned} \Vert u_{\gamma '} - u_\gamma \Vert _{L^2(Q)} \le L|\gamma ' - \gamma |^{\frac{1}{2}}. \end{aligned}$$
(6.4)

Proof

The first part of the theorem follows from Remark 6.1. We only have to prove (6.4). For every \(\gamma '\) we define

$$\begin{aligned} {\hat{u}}_{\gamma '} = \left\{ \begin{array}{cl} u_\gamma &{}\text {if } \gamma< \gamma ',\\ \frac{\gamma '}{\gamma }u_\gamma &{}\text {if } \gamma> \gamma ',\end{array}\right. \quad \text { and }\quad {\hat{v}}_{\gamma '} = \left\{ \begin{array}{cl} u_{\gamma '}&{}\text {if } \gamma ' < \gamma ,\\ \frac{\gamma }{\gamma '}u_{\gamma '}&{}\text {if } \gamma ' > \gamma .\end{array}\right. \end{aligned}$$
(6.5)

Then we have

$$\begin{aligned} {\hat{u}}_{\gamma '}, {\hat{v}}_{\gamma '} \in {U_\gamma }\cap {U_{\gamma '}},\ \ {\hat{u}}_{\gamma '} \rightarrow u_\gamma \text { in } L^\infty (Q) \text { and } {\hat{v}}_{\gamma '} \rightarrow u_\gamma \text { in } L^r(0,T;L^2(\varOmega )).\nonumber \\ \end{aligned}$$
(6.6)

From here we infer that \(v_{\gamma '} \in {U_\gamma }\cap B_\varepsilon (u_\gamma )\) for \(\gamma '\) close enough to \(\gamma \) with \(B_\varepsilon (u_\gamma )\) defined in (5.3). Therefore, we get

$$\begin{aligned} \frac{\delta }{2}\Vert {\hat{v}}_{\gamma '} - u_\gamma \Vert ^2_{L^2(Q)} \le J({\hat{v}}_{\gamma '}) - J(u_\gamma ). \end{aligned}$$
(6.7)

In the case \(\gamma ' < \gamma \), using (6.7), the optimality of \(u_{\gamma '}\), and the definition of \({\hat{v}}_{\gamma '}\) we obtain with the mean value theorem

$$\begin{aligned}&\Vert u_{\gamma '} - u_\gamma \Vert ^2_{L^2(Q)} \le \frac{2}{\delta }\big [(J(u_{\gamma '}) - J({\hat{u}}_{\gamma '})) + (J({\hat{u}}_{\gamma '}) - J(u_\gamma ))\big ]\\&\le \frac{2}{\delta }(J({\hat{u}}_{\gamma '}) - J(u_\gamma )) \le C_1\Vert {\hat{u}}_{\gamma '} - u_\gamma \Vert _{L^\infty (Q)} = \frac{C_1}{\gamma }\Vert u_\gamma \Vert _{L^\infty (Q)}|\gamma ' - \gamma |. \end{aligned}$$

In the case \(\gamma ' > \gamma \) we proceed as follows

$$\begin{aligned}&\Vert {\hat{v}}_{\gamma '} - u_\gamma \Vert ^2_{L^2(Q)} \le \frac{2}{\delta }\big [(J({\hat{v}}_{\gamma '}) - J(u_{\gamma '})) + (J(u_{\gamma '}) - J(u_\gamma ))\big ]\\&\le \frac{2}{\delta }\big (J({\hat{v}}_{\gamma '}) - J(u_{\gamma '})\big ) \le C_2\Vert {\hat{v}}_{\gamma '} - u_{\gamma '}\Vert _{L^r(0,T;L^2(\varOmega ))}\\&= \frac{C_2}{\gamma '}\Vert u_{\gamma '}\Vert _{L^r(0,T;L^2(\varOmega ))}|\gamma ' - \gamma | \le C_3|\gamma ' - \gamma |. \end{aligned}$$

From here we get

$$\begin{aligned}&\Vert u_{\gamma '} - u_\gamma \Vert _{L^2(Q)} \le \Vert u_{\gamma '} - {\hat{v}}_{\gamma '}\Vert _{L^2(Q)} + \Vert {\hat{v}}_{\gamma '} - u_{\gamma }\Vert _{L^2(Q)}\\&\le \frac{|\gamma ' - \gamma |}{\gamma '}\Vert u_{\gamma '}\Vert _{L^2(Q)} + \sqrt{C_3}|\gamma ' - \gamma |^{\frac{1}{2}} \le C_4|\gamma ' - \gamma |^{\frac{1}{2}}, \end{aligned}$$

which concludes the proof. \(\square \)

Theorems 5.2 and 6.2 imply Hölder stability with respect to \(\gamma \) of the optimal controls if the sufficient second order condition \(J''(u_\gamma )v^2 > 0\) \(\forall v \in C_{{\bar{u}}} \setminus \{0\}\) holds. Now, we are interested in proving Lipschitz stability. To this end we need to make a stronger assumption, namely

$$\begin{aligned} J''(u_{{{\bar{\gamma }}}})v^2> 0\quad \forall v \in L^2(Q) \setminus \{0\}, \quad y_0 \in C_0(\varOmega ), \quad \text {and}\quad {{\hat{r}}}> \frac{4}{4-n}, \end{aligned}$$
(6.8)

where \({{\bar{\gamma }}}> 0\) is fixed and \(C_0(\varOmega )\) denotes the space of continuous real valued functions on \({\bar{\varOmega }}\) vanishing on \(\Gamma \). From the first assumption in (6.8) we deduce the existence of strictly positive numbers \(\rho \) and \(\nu \) such that

$$\begin{aligned} J''(u)v^2 \ge \nu \Vert v\Vert ^2_{L^2(Q)}\quad \forall v \in L^2(Q)\ \text { and } \ \forall u \in B_\rho (u_{{{\bar{\gamma }}}}), \end{aligned}$$
(6.9)

where \(B_\rho (u_{{{\bar{\gamma }}}})\) denotes the \(L^{{\hat{r}}}(0,T;L^2(\varOmega ))\) closed ball. Indeed, if (6.9) does not hold, then we can take sequences \(\{u_k\}_{k = 1}^\infty \subset L^{{\hat{r}}}(0,T;L^2(\varOmega ))\) and \(\{v_k\}_{k = 1}^\infty \subset L^2(Q)\) satisfying

$$\begin{aligned} \lim _{k \rightarrow \infty }\Vert u_k - u_{{{\bar{\gamma }}}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} = 0, \Vert v_k\Vert _{L^2(Q)} = 1, v_k \rightharpoonup v \text { in } L^2(Q), J''(u_k)v_k^2 \le \frac{1}{k}. \end{aligned}$$

It is easy to pass to the limit and to deduce

$$\begin{aligned} J''(u_{{{\bar{\gamma }}}})v^2 \le \liminf _{k \rightarrow \infty }J''(u_k)v_k^2 \le 0. \end{aligned}$$

This inequality and (6.8) yield \(v = 0\). But, arguing as in the proof of Theorem 5.3 we infer

$$\begin{aligned} \kappa = \liminf _{k \rightarrow \infty }\kappa \Vert v_k\Vert ^2_{L^2(Q)} = \liminf _{k \rightarrow \infty }J''(u_k)v_k^2 \le 0, \end{aligned}$$

which contradicts our assumption \(\kappa > 0\).

We finish this section by proving the next theorem.

Theorem 6.3

Let \(u_{{{\bar{\gamma }}}}\) be a local minimizer of (P\(_{{{\bar{\gamma }}}}\)). We assume that (6.8) holds and that \(\rho \) satisfies (6.9). Then, there exists \({\bar{\varepsilon }}\in (0,{\bar{\gamma }})\) such that \( \text{(P }_\gamma ) \) has a unique local minimizer \(u_\gamma \) in the interior of the \(L^{{\hat{r}}}(0,T;L^2(\varOmega ))\) ball \(B_\rho (u_{{{\bar{\gamma }}}})\) for every \(\gamma \in ({{\bar{\gamma }}}- {\bar{\varepsilon }},{{\bar{\gamma }}}+ {\bar{\varepsilon }})\). Moreover, there exists a constant L such that

$$\begin{aligned} \Vert u_\gamma - u_{{{\bar{\gamma }}}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} \le L|\gamma - {{\bar{\gamma }}}|\quad \forall \gamma \in ({{\bar{\gamma }}}- {\bar{\varepsilon }},{{\bar{\gamma }}}+ {\bar{\varepsilon }}). \end{aligned}$$
(6.10)

Proof

Let us take \(\rho > 0\) such that (6.9) holds. Then, J has at most one local (and global) minimizer \(u_\gamma \) in the closed set \(B_\rho (u_{{{\bar{\gamma }}}}) \cap {U_{ad}}\). This is a consequence of the strict convexity of J in the ball \(B_\rho (u_{{{\bar{\gamma }}}})\); see (6.9). We will prove that this local minimizer belongs to the interior of the \(L^{{\hat{r}}}(0,T;L^2(\varOmega ))\) ball \(B_\rho (u_{{{\bar{\gamma }}}})\) if \(\gamma \) is close enough to \({{\bar{\gamma }}}\), and consequently it is a local minimizer of \( \text{(P }_\gamma ) \). In order to prove this, as well as (6.10), we reformulate the control problem \( \text{(P }_\gamma ) \) as follows

$$\begin{aligned} \text{(Q }_\gamma ) \quad \inf _{u \in {K_{ad}}} J_\gamma (u):= \frac{1}{2}\int _Q (y_{\gamma ,u}(x,t) - y_d(x))^2\,\mathrm {d}x\,\mathrm {d}t+ \frac{\kappa \gamma ^2}{2}\int _Q u(x,t)^2\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$

where

$$\begin{aligned} {K_{ad}}= \{u \in L^{{\hat{r}}}(0,T;L^2(\varOmega )) : \Vert u(t)\Vert _{L^1(\varOmega )} \le 1 \text { for a.a. } t \in (0,T)\} \end{aligned}$$

and \(y_{\gamma ,u}\) is the solution of the semilinear parabolic equation

$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle \frac{\partial y}{\partial t} + Ay + a(x,t,y) = \gamma u &{} \text{ in } Q = \varOmega \times (0,T),\\ y = 0 \text{ on } \varSigma = \Gamma \times (0,T),&{} y(0) = y_0 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$
(6.11)

It is obvious that the problems \( \text{(P }_\gamma ) \) and \( \text{(Q }_\gamma ) \) are equivalent for every \(\gamma \). This equivalence is understood in the sense that u is a local (global) minimizer of \( \text{(Q }_\gamma ) \) if and only if \(u_\gamma = \gamma u\) is a local (global) minimizer of \( \text{(P }_\gamma ) \), and \(J(u_\gamma ) = J_\gamma (u)\); recall Remark 4.2.

Take \(\varepsilon \in (0,{{\bar{\gamma }}})\) and \({\bar{\rho }} \in (0,\rho ]\) such that \(({{\bar{\gamma }}}+ \varepsilon ){\bar{\rho }} + \varepsilon \Vert {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} < \rho \). Then, we have with the notation \(u_{{\bar{\gamma }}}= {{\bar{\gamma }}}{\bar{u}}\) and \(u_\gamma = \gamma u\)

$$\begin{aligned}&\Vert u_\gamma - u_{{\bar{\gamma }}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega )} \le \gamma \Vert u - {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} + |\gamma - {{\bar{\gamma }}}|\Vert {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))}\\&\le ({{\bar{\gamma }}}+ \varepsilon ){\bar{\rho }} + \varepsilon \Vert u_{{\bar{\gamma }}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} < \rho \ \ \forall u \in B_{{\bar{\rho }}}({\bar{u}}) \text { and } \forall \gamma \in ({{\bar{\gamma }}}- \varepsilon ,{{\bar{\gamma }}}+ \varepsilon ). \end{aligned}$$

Due to (6.9) and the fact that \(J_\gamma ''(u)v^2 = \gamma ^2 J''(u_\gamma )v^2\), we deduce that

$$\begin{aligned} J''_\gamma (u)v^2 \ge \gamma ^2\nu \Vert v\Vert ^2_{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} \ge ({{\bar{\gamma }}}- \varepsilon )^2\nu \Vert v\Vert ^2_{L^{{\hat{r}}}(0,T;L^2(\varOmega ))}\quad \forall u \in B_{{\bar{\rho }}}({\bar{u}}). \end{aligned}$$

Therefore, \(J_\gamma \) is strictly convex on the ball \(B_{{\bar{\rho }}}({\bar{u}})\). Hence, a control u is a local solution of \( \text{(Q }_\gamma ) \) in the interior of \(B_{{\bar{\rho }}}({\bar{u}})\) if and only if u satisfies the optimality system

$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle \frac{\partial y}{\partial t} + Ay + a(x,t,y) = \gamma u \ \text{ in } Q,\\ y = 0 \text{ on } \varSigma ,\ \ y(0) = y_0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(6.12)
$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle -\frac{\partial \varphi }{\partial t} + A^*\varphi + \frac{\partial a}{\partial y}(x,t,y)\varphi = y - y_d\ \text{ in } Q,\\ \varphi = 0 \text{ on } \varSigma ,\ \ \varphi (T) = 0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(6.13)
$$\begin{aligned}&\int _Q\mu (v - u)\,\mathrm {d}x\,\mathrm {d}t\le 0\quad \forall v \in {K_{ad}}, \end{aligned}$$
(6.14)
$$\begin{aligned}&\gamma \varphi + \kappa \gamma ^2 u + \mu = 0. \end{aligned}$$
(6.15)

Denote by \({\bar{y}}\) and \({\bar{\varphi }}\) the state and adjoint state associated to \({\bar{u}}\). Our goal is to apply [10, Theorem 2.4] to the previous optimality system. To this end we define the spaces:

$$\begin{aligned}&{\mathcal {Y}} = \{y \in W(0,T) \cap C({\bar{Q}}) : \frac{\partial y}{\partial t} + Ay \in L^{{\hat{r}}}(0,T;L^2(\varOmega ))\},\\&\varPhi = \{\varphi \in H^1(Q) \cap C({\bar{Q}}) : -\frac{\partial \varphi }{\partial t} + A^*\varphi \in L^{{\hat{r}}}(0,T;L^2(\varOmega )) \text { and } \varphi (T) = 0\},\\&X = {\mathcal {Y}} \times \varPhi \times L^{{\hat{r}}}(0,T;L^2(\varOmega )),\ \ Y = {\mathbb {R}},\ \ Z = C_0(\varOmega ) \times L^{{\hat{r}}}(0,T;L^2(\varOmega ))^3. \end{aligned}$$

On \({\mathcal {Y}}\) and \(\varPhi \) we consider the graph norms

$$\begin{aligned}&\Vert y\Vert _{{\mathcal {Y}}} = \Vert y\Vert _{W(0,T)} + \Vert y\Vert _{C({\bar{Q}})} + \Big \Vert \frac{\partial y}{\partial t} + Ay\Big \Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))},\\&\Vert \varphi \Vert _\varPhi = \Vert \varphi \Vert _{H^1(Q)} + \Vert \varphi \Vert _{C({\bar{Q}})} + \Big \Vert -\frac{\partial \varphi }{\partial t} + A^*\varphi \Big \Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))}. \end{aligned}$$

Thus, X is a Banach space. Moreover, we introduce the mapping \(f:X \times Y \longrightarrow Z\) and the multivalued function \(F:X \longrightarrow Z\)

$$\begin{aligned} f((y,\varphi ,u),\gamma )= & {} \left( \begin{array}{c}y(0) - y_0\\ \displaystyle \frac{\partial y}{\partial t} + Ay + a(\cdot ,\cdot ,y) - \gamma u\\ \displaystyle -\frac{\partial \varphi }{\partial t} + A^*\varphi + \frac{\partial a}{\partial y}(\cdot ,\cdot ,y)\varphi - y + y_d\\ \gamma \varphi + \gamma ^2\kappa u\end{array}\right) , \\ F(y,\varphi ,u)= & {} \left( \begin{array}{c} 0\\ 0\\ 0\\ F_0(u)\end{array}\right) , \end{aligned}$$

where the multivalued function \(F_0:L^{{\hat{r}}}(0,T;L^2(\varOmega )) \longrightarrow L^{{\hat{r}}}(0,T;L^2(\varOmega ))\) is defined by

$$\begin{aligned} F_0(u) = \left\{ \begin{array}{cl} \emptyset &{}\text {if } u \not \in {K_{ad}},\\ \displaystyle \Big \{\mu \in L^{{\hat{r}}}(0,T;L^2(\varOmega )) : \int _Q\mu (v - u)\,\mathrm {d}x\,\mathrm {d}t\le 0\ \forall v \in {K_{ad}}\Big \},&{}\text {otherwise.}\end{array}\right. \end{aligned}$$

Due to the regularity \(y_0 \in C_0(\varOmega )\), see assumption (6.8), we deduce from (6.12) that \({\bar{y}} \in {\mathcal {Y}}\). Therefore, we have that \(({\bar{y}},{\bar{\varphi }},{\bar{u}}) \in X\). Moreover, \(({\bar{y}},{\bar{\varphi }},{\bar{u}})\) satisfies the optimality system (6.12)–(6.15), which implies that \(0 \in f(({\bar{y}},{\bar{\varphi }},{\bar{u}}),{{\bar{\gamma }}}) + F({\bar{y}},{\bar{\varphi }},{\bar{u}})\). Using our assumptions on a and the continuous embedding \({\mathcal {Y}} \subset C({\bar{Q}})\) we deduce that the function f is of class \(C^1\). Then, the function \(g:X \longrightarrow Z\), defined by

$$\begin{aligned} g(y,\varphi ,u) = f(({\bar{y}},{\bar{\varphi }},{\bar{u}}),{{\bar{\gamma }}}) + D_{(y,\varphi ,u)}f(({\bar{y}},{\bar{\varphi }},{\bar{u}}),{{\bar{\gamma }}})(y - {\bar{y}},\varphi - {\bar{\varphi }},u - {\bar{u}}), \end{aligned}$$

strongly approximates f at \((({\bar{y}},{\bar{\varphi }},{\bar{u}}),{{\bar{\gamma }}})\), and \(g({\bar{y}},{\bar{\varphi }},{\bar{u}}) = f(({\bar{y}},{\bar{\varphi }},{\bar{u}}),{{\bar{\gamma }}})\); see [19] for the definition of a strong approximation.

We will apply [10, Theorem 2.4] to deduce the existence of \({\bar{\varepsilon }} \in (0,\varepsilon ]\) and \({\tilde{\rho }} \in (0,{\bar{\rho }}]\) such that (6.12)–(6.15) has a unique solution u in the interior of the ball \(B_{{\tilde{\rho }}}({\bar{u}})\) for every \(\gamma \in ({{\bar{\gamma }}}- {\bar{\varepsilon }},{{\bar{\gamma }}}+ {\bar{\varepsilon }})\). Moreover, these solutions satisfy

$$\begin{aligned} \Vert u - {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} \le \lambda |\gamma - {{\bar{\gamma }}}|. \end{aligned}$$
(6.16)

for some \(\lambda > 0\). For this purpose it is enough to prove that the equation

$$\begin{aligned} \beta \in g(y,\varphi ,u) + F(y,\varphi ,u) \end{aligned}$$
(6.17)

has a unique solution \((y_\beta ,\varphi _\beta ,u_\beta ) \in X\) for every \(\beta = (\beta _i)_{i = 1}^4 \in Z\) and the Lipschitz property

$$\begin{aligned} \Vert (y_{{\hat{\beta }}},\varphi _{{\hat{\beta }}},u_{{\hat{\beta }}}) - (y_\beta ,\varphi _\beta ,u_\beta )\Vert _X \le \lambda \Vert {\hat{\beta }} - \beta \Vert _Z \end{aligned}$$
(6.18)

holds for some \(\lambda > 0\) and all \({\hat{\beta }}, \beta \in Z\). First, we prove the existence of a unique solution. To this end we consider the optimal control problem

$$\begin{aligned} (\mathrm{P}_\beta )\quad \inf _{u \in {K_{ad}}} {\mathcal {J}}_\beta (u), \end{aligned}$$

where

$$\begin{aligned} {\mathcal {J}}_\beta (u):=&\frac{1}{2}\int _Q \big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]y^2\,\mathrm {d}x\,\mathrm {d}t+ \frac{\kappa {{\bar{\gamma }}}^2}{2}\int _Q u(x,t)^2\,\mathrm {d}x\,\mathrm {d}t\\&+ \int _Q\beta _3 y\,\mathrm {d}x\,\mathrm {d}t+ \int _Q({{\bar{\gamma }}}{\bar{\varphi }} + {{\bar{\gamma }}}^2\kappa {\bar{u}} - \beta _4)u\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$

and y satisfies the equation

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \frac{\partial y}{\partial t} + A y + \frac{\partial a}{\partial y}(x,t,{\bar{y}})y = {{\bar{\gamma }}}u + \beta _2\ \text{ in } Q,\\ y = 0 \text{ on } \varSigma ,\ \ y(0) = \beta _1 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$
(6.19)

Let us consider the solution \(\xi _\beta \in {\mathcal {Y}}\) of the equation

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \frac{\partial \xi }{\partial t} + A\xi + \frac{\partial a}{\partial y}(x,t,{\bar{y}})\xi = \beta _2\ \text{ in } Q,\\ \xi = 0 \text{ on } \varSigma ,\ \ \xi (0) = \beta _1 \text{ in } \varOmega .\end{array}\right. \end{aligned}$$
(6.20)

According to (2.17) we have that \(y = {{\bar{\gamma }}}G'(u_{{{\bar{\gamma }}}})u + \xi _\beta = {{\bar{\gamma }}}z_u + \xi _\beta \). Inserting this identity in the cost functional we get

$$\begin{aligned} {\mathcal {J}}_\beta (u)=&\frac{{{\bar{\gamma }}}^2}{2}\left\{ \int _Q \big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]z_u^2\,\mathrm {d}x\,\mathrm {d}t+ \kappa \int _Q u^2\,\mathrm {d}x\,\mathrm {d}t\right\} \\&+ {{\bar{\gamma }}}\int _Q\Big (\big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]\xi _\beta + \beta _3\Big )z_u\,\mathrm {d}x\,\mathrm {d}t\\&+ \int _Q({{\bar{\gamma }}}{\bar{\varphi }} + {{\bar{\gamma }}}^2\kappa u_{{{\bar{\gamma }}}} - \beta _4)u\,\mathrm {d}x\,\mathrm {d}t\\&+ \int _Q\Big (\frac{1}{2}\big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]\xi _\beta ^2 + \beta _3\xi _\beta \Big )\,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$

From (2.21), (6.9), and the continuity of the mapping \(u \rightarrow z_u\) in \(L^2(Q)\) we deduce the existence of two constants \(C_1\) and \(C_2\) such that

$$\begin{aligned} {\mathcal {J}}_\beta (u) \ge \frac{{{\bar{\gamma }}}^2}{2}\nu \Vert u\Vert ^2_{L^2(Q)} + C_1\Vert u\Vert _{L^2(Q)} + C_2. \end{aligned}$$

Therefore, \({\mathcal {J}}_\beta \) is a coercive, continuous, and strictly convex quadratic functional on \(L^2(Q)\). As a consequence, we infer the existence and uniqueness of a minimizer \({\tilde{u}}_\beta \) of \({\mathcal {J}}_\beta \) on the set

$$\begin{aligned} \tilde{K}_{ad} = \{u \in L^2(Q) : \Vert u(t)\Vert _{L^1(\varOmega )} \le 1 \text { for a.a. } t \in (0,T)\}. \end{aligned}$$

Similarly as in Theorem 3.1, we deduce the existence of elements \({\tilde{y}}_\beta \in W(0,T)\), \({\tilde{\varphi }}_\beta \in H^1(Q)\), and \({\tilde{\mu }}_\beta \in L^2(Q)\) satisfying

$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle \frac{\partial {\tilde{y}}_\beta }{\partial t} + A{\tilde{y}}_\beta + \frac{\partial a}{\partial y}(x,t,{\bar{y}}){\tilde{y}}_\beta = {{\bar{\gamma }}}{\tilde{u}}_\beta + \beta _2\ \text{ in } Q,\\ {\tilde{y}}_\beta = 0 \text{ on } \varSigma ,\ \ {\tilde{y}}_\beta (0) = \beta _1 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(6.21)
$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle -\frac{\partial {\tilde{\varphi }}_\beta }{\partial t} + A^*{\tilde{\varphi }}_\beta + \frac{\partial a}{\partial y}(x,t,{\bar{y}}){\tilde{\varphi }}_\beta = \big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]{\tilde{y}}_\beta + \beta _3\ \text{ in } Q,\\ {\tilde{\varphi }}_\beta = 0 \text{ on } \varSigma ,\ \ {\tilde{\varphi }}_\beta (T) = 0 \text{ in } \varOmega ,\end{array}\right. \end{aligned}$$
(6.22)
$$\begin{aligned}&\int _Q{\tilde{\mu }}_\beta (u - {\tilde{u}}_\beta )\,\mathrm {d}x\,\mathrm {d}t\le 0\quad \forall u \in \tilde{K}_{ad}, \end{aligned}$$
(6.23)
$$\begin{aligned}&{{\bar{\gamma }}}{\tilde{\varphi }}_\beta + {{\bar{\gamma }}}^2\kappa {\tilde{u}}_\beta + {{\bar{\gamma }}}{\bar{\varphi }} + {{\bar{\gamma }}}^2\kappa {\bar{u}} - \beta _4+ {\tilde{\mu }}_\beta = 0. \end{aligned}$$
(6.24)

Arguing similarly as in the proof of Theorem 4.4 we deduce that \({\tilde{u}}_\beta \) and \({\tilde{\mu }}_\beta \) belong to the space \(L^{{\hat{r}}}(0,T;L^2(\varOmega ))\). Thus, \({\tilde{u}}_\beta \) is the unique solution of (P\(_\beta \)). Moreover, from (6.21) and (6.22) along with (6.8) we infer that \({\tilde{y}}_\beta \in {\mathcal {Y}}\) and \({\tilde{\varphi }}_\beta \in \varPhi \). Hence, we have that \(({\tilde{y}}_\beta ,{\tilde{\varphi }}_\beta ,{\tilde{u}}_\beta ) \in X\) and (6.23) holds for every \(u \in {K_{ad}}\). Due to the convexity of (P\(_\beta \)), we know that (6.21)–(6.24) are necessary and sufficient conditions of optimality for (P\(_\beta \)). This fact and the strict convexity of \({\mathcal {J}}_\beta \) imply that the system (6.21)–(6.24) has a unique solution \(({\tilde{y}}_\beta ,{\tilde{\varphi }}_\beta ,{\tilde{u}}_\beta ,{\tilde{\mu }}_\beta )\). If we set \(y_\beta = {\tilde{y}}_\beta + {\bar{y}}\), \(\varphi _\beta = {\tilde{\varphi }}_\beta + {\bar{\varphi }}\), \(u_\beta = {\tilde{u}}_\beta + {\bar{u}}\), and \(\mu _\beta = {\tilde{\mu }}_\beta \), (6.21)–(6.24) yields that \((y_\beta ,\varphi _\beta ,u_\beta )\) is the unique element of X satisfying (6.17).

Now, we prove that this solution is Lipschitz with respect to \(\beta \). First, we observe that (6.24) can be written as

$$\begin{aligned} {{\bar{\gamma }}}\varphi _\beta + {{\bar{\gamma }}}^2\kappa u_\beta - \beta _4 + \mu _\beta = 0. \end{aligned}$$
(6.25)

Given \(\beta , {\hat{\beta }} \in Z\), we infer from (6.23)-(6.24) and (6.25) for \(\beta \) and \({\hat{\beta }}\)

$$\begin{aligned}&\int _\varOmega ({{\bar{\gamma }}}\varphi _\beta (t) + {{\bar{\gamma }}}^2\kappa u_\beta (t) - \beta _4(t))(u_{{\hat{\beta }}}(t) - u_\beta (t))\,\mathrm {d}x\,\mathrm {d}t\le 0,\\&\int _\varOmega ({{\bar{\gamma }}}\varphi _{{\hat{\beta }}}(t) + {{\bar{\gamma }}}^2\kappa u_{{\hat{\beta }}}(t) - {\hat{\beta }}_4(t))(u_\beta (t) - u_{{\hat{\beta }}}(t))\,\mathrm {d}x\,\mathrm {d}t\le 0. \end{aligned}$$

Adding these inequalities we get

$$\begin{aligned} {{\bar{\gamma }}}^2\kappa \Vert u_{{\hat{\beta }}}(t)&- u_\beta (t)\Vert ^2_{L^2(\varOmega )} \le {{\bar{\gamma }}}\int _\varOmega (\varphi _\beta (t) - \varphi _{{\hat{\beta }}}(t))(u_{{\hat{\beta }}}(t) - u_\beta (t))\,\mathrm {d}x\nonumber \\&+ \Vert {\hat{\beta }}_4(t) - \beta _4(t)\Vert _{L^2(\varOmega )}\Vert u_{{\hat{\beta }}}(t) - u_\beta (t)\Vert _{L^2(\varOmega )} \end{aligned}$$
(6.26)

for almost every \(t \in (0,T)\). Now, taking into account that \(y_{{\hat{\beta }}} - y_\beta = {\tilde{y}}_{{\hat{\beta }}} - {\tilde{y}}_\beta \), \(\varphi _{{\hat{\beta }}} - \varphi _\beta = {\tilde{\varphi }}_{{\hat{\beta }}} - {\tilde{\varphi }}_\beta \), and \(u_{{\hat{\beta }}} - u_\beta = {\tilde{u}}_{{\hat{\beta }}} - {\tilde{u}}_\beta \), subtracting the equations (6.21) satisfied by \(y_{{\hat{\beta }}}\) and \(y_\beta \), and the equations (6.22) for \(\varphi _{{\hat{\beta }}}\) and \(\varphi _\beta \), respectively, we obtain

$$\begin{aligned}&\ \ {{\bar{\gamma }}}\int _Q(\varphi _\beta (t) - \varphi _{{\hat{\beta }}}(t))(u_{{\hat{\beta }}}(t) - u_\beta (t))\,\mathrm {d}x\\&= \int _Q\left\{ \Big (\frac{\partial }{\partial t} + A + \frac{\partial a}{\partial y}(x,t,{\bar{y}})\Big )(y_{{\hat{\beta }}} - y_\beta )(\varphi _\beta - \varphi _{{\hat{\beta }}}) - ({\hat{\beta }}_2 - \beta _2)(\varphi _\beta - \varphi _{{\hat{\beta }}})\right\} \,\mathrm {d}x\,\mathrm {d}t\\&= \int _Q\left\{ \Big (-\frac{\partial }{\partial t} + A^* + \frac{\partial a}{\partial y}(x,t,{\bar{y}})\Big )(\varphi _\beta - \varphi _{{\hat{\beta }}})(y_{{\hat{\beta }}} - y_\beta ) - ({\hat{\beta }}_2 - \beta _2)(\varphi _\beta - \varphi _{{\hat{\beta }}})\right\} \,\mathrm {d}x\,\mathrm {d}t\\&- \int _\varOmega ({\hat{\beta }}_1 - \beta _1)(\varphi _\beta (0) - \varphi _{{\hat{\beta }}}(0))\,dx\\&= -\int _Q\Big \{\big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ](y_{{\hat{\beta }}} - y_\beta )^2 + ({\hat{\beta }}_2 - \beta _2)(\varphi _\beta - \varphi _{{\hat{\beta }}})\Big \}\,\mathrm {d}x\,\mathrm {d}t\\&-\int _Q({\hat{\beta }}_3 - \beta _3)(y_{{\hat{\beta }}} - y_\beta )\,\mathrm {d}x\,\mathrm {d}t- \int _\varOmega ({\hat{\beta }}_1 - \beta _1)(\varphi _\beta (0) - \varphi _{{\hat{\beta }}}(0))\,\mathrm {d}x. \end{aligned}$$

Let us denote by \(\xi _\beta \) and \(\xi _{{\hat{\beta }}}\) the solutions of (6.20) corresponding to \((\beta _1,\beta _2)\) and \(({\hat{\beta }}_1,{\hat{\beta }}_2)\), respectively. Then, we have that \(y_{{\hat{\beta }}} - y_\beta = {{\bar{\gamma }}}G'({\bar{u}})(u_{{\hat{\beta }}} - u_\beta ) + \xi _{{\hat{\beta }}} - \xi _\beta = {{\bar{\gamma }}}z_{u_{{\hat{\beta }}} - u_\beta } + (\xi _{{\hat{\beta }}} - \xi _\beta )\). Inserting this identity in the above equality we infer

$$\begin{aligned}&{{\bar{\gamma }}}\int _Q(\varphi _\beta (t) - \varphi _{{\hat{\beta }}}(t))(u_{{\hat{\beta }}}(t) - u_\beta (t))\,\mathrm {d}x\\&\quad = -{{\bar{\gamma }}}^2\int _Q\big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]z^2_{u_{{\hat{\beta }}} - u_\beta }\,\mathrm {d}x\,\mathrm {d}t\\&\qquad - \int _Q\big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ][(\xi _{{\hat{\beta }}} - \xi _\beta )^2 + 2z_{u_{{\hat{\beta }}} - u_\beta }(\xi _{{\hat{\beta }}} - \xi _\beta )]\,\mathrm {d}x\,\mathrm {d}t\\&\qquad - \int _Q\Big \{({\hat{\beta }}_2 - \beta _2)(\varphi _{{\hat{\beta }}} - \varphi _\beta ) + ({\hat{\beta }}_3 - \beta _3)(y_{{\hat{\beta }}} - y_\beta )\Big \}\,\mathrm {d}x\,\mathrm {d}t\\&\qquad - \int _\varOmega ({\hat{\beta }}_1 - \beta _1)(\varphi _\beta (0) - \varphi _{{\hat{\beta }}}(0))\,\mathrm {d}x\\&\quad \le -{{\bar{\gamma }}}^2\int _Q\big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]z^2_{u_{{\hat{\beta }}} - u_\beta }\,\mathrm {d}x\,\mathrm {d}t\\&\qquad + C_3\Big \{\Vert {\hat{\beta }}_1 - \beta _1\Vert ^2_{L^2(\varOmega )} + \Vert {\hat{\beta }}_2 - \beta _2\Vert ^2_{L^2(Q)}\\&\qquad + \Vert u_{{\hat{\beta }}} - u_\beta \Vert _{L^2(Q)}\big [\Vert {\hat{\beta }}_1 - \beta _1\Vert _{L^2(\varOmega )} + \Vert {\hat{\beta }}_2 - \beta _2\Vert _{L^2(Q)}\big ]\\&\qquad + \Vert {\hat{\beta }}_2 - \beta _2\Vert _{L^2(Q)}\Vert \varphi _{{\hat{\beta }}} - \varphi _\beta \Vert _{L^2(Q)} + \Vert {\hat{\beta }}_3 - \beta _3\Vert _{L^2(Q)}\Vert y_{{\hat{\beta }}} - y_\beta \Vert _{L^2(Q)}\\&\qquad + \Vert {\hat{\beta }}_1 - \beta _1\Vert _{L^2(\varOmega )}\Vert \varphi _\beta (0) - \varphi _{{\hat{\beta }}}(0)\Vert _{L^2(\varOmega )}\Big \}. \end{aligned}$$

Now, from the equations satisfied by \(y_{{\hat{\beta }}} - y_\beta \) and \(\varphi _{{\hat{\beta }}} - \varphi _\beta \) we get

$$\begin{aligned}&\Vert y_{{\hat{\beta }}} - y_\beta \Vert _{W(0,T)} \le C_4\Big (\Vert u_{{\hat{\beta }}} - u_\beta \Vert _{L^2(Q)} + \Vert {\hat{\beta }}_2 - \beta _2\Vert _{L^2(Q)} + \Vert {\hat{\beta }}_1 - \beta _1\Vert _{L^2(\varOmega )}\Big ), \end{aligned}$$
(6.27)
$$\begin{aligned}&\Vert \varphi _{{\hat{\beta }}} - \varphi _\beta \Vert _{H^1(Q)} \le C_5\Big (\Vert y_{{\hat{\beta }}} - y_\beta \Vert _{L^2(Q)} + \Vert {\hat{\beta }}_3 - \beta _3\Vert _{L^2(Q)}\Big ). \end{aligned}$$
(6.28)

Using the continuous embeddings \(W(0,T) \subset L^2(Q)\) and \(H^1(Q) \subset C([0,T];L^2(\varOmega ))\), and the estimates (6.27) and (6.28), we infer

$$\begin{aligned}&{{\bar{\gamma }}}\int _Q(\varphi _\beta (t) - \varphi _{{\hat{\beta }}}(t))(u_{{\hat{\beta }}}(t) - u_\beta (t))\,\mathrm {d}x\\&\quad \le -{{\bar{\gamma }}}^2\int _Q\big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]z^2_{u_{{\hat{\beta }}} - u_\beta }\,\mathrm {d}x\,\mathrm {d}t\\&\qquad + C_6\Big \{\Vert u_{{\hat{\beta }}} - u_\beta \Vert _{L^2(Q)}\big [\Vert {\hat{\beta }}_1 - \beta _1\Vert _{L^2(\varOmega )} + \Vert {\hat{\beta }}_2 - \beta _2\Vert _{L^2(Q)} + \Vert {\hat{\beta }}_3 - \beta _3\Vert _{L^2(Q)}\big ]\\&\qquad + \Vert {\hat{\beta }}_1 - \beta _1\Vert ^2_{L^2(\varOmega )} + \Vert {\hat{\beta }}_2 - \beta _2\Vert ^2_{L^2(Q)} + \Vert {\hat{\beta }}_3 - \beta _3\Vert ^2_{L^2(Q)}\Big \}. \end{aligned}$$

Combining this inequality with (6.26) and using (6.9) we deduce

$$\begin{aligned}&{{\bar{\gamma }}}^2\nu \Vert u_{{\hat{\beta }}} - u_\beta \Vert ^2_{L^2(Q)} \le {{\bar{\gamma }}}^2J''({\bar{u}})(u_{{\hat{\beta }}} - u_\beta )^2\\&\quad = {{\bar{\gamma }}}^2\Big \{\int _Q\big [1 - \frac{\partial ^2a}{\partial y^2}(x,t,{\bar{y}}){\bar{\varphi }}\big ]z^2_{u_{{\hat{\beta }}} - u_\beta }\,\mathrm {d}x\,\mathrm {d}t+ \kappa \Vert u_{{\hat{\beta }}} - u_\beta \Vert ^2_{L^2(Q)}\Big \}\\&\quad \le C_7\Big \{\Vert u_{{\hat{\beta }}} - u_\beta \Vert _{L^2(Q)}\Big (\Vert {\hat{\beta }}_1 - \beta _1\Vert _{L^2(\varOmega )} + \sum _{j = 2}^4\Vert {\hat{\beta _j}} - \beta _j\Vert _{L^2(Q)}\Big )\\&\qquad + \Vert {\hat{\beta }}_1 - \beta _1\Vert ^2_{L^2(\varOmega )} + \sum _{j = 2}^3\Vert {\hat{\beta _j}} - \beta _j\Vert ^2_{L^2(Q)} \Big \}. \end{aligned}$$

This yields

$$\begin{aligned} \Vert u_{{\hat{\beta }}} - u_\beta \Vert _{L^2(Q)} \le C_8\Vert {\hat{\beta }} - \beta \Vert _Z. \end{aligned}$$
(6.29)

Using (6.27) and (6.29) it follows that

$$\begin{aligned} \Vert y_{{\hat{\beta }}} - y_\beta \Vert _{W(0,T)} \le C_9\Vert {\hat{\beta }} - \beta \Vert _Z. \end{aligned}$$
(6.30)

Now, (6.28) and (6.30) lead to

$$\begin{aligned} \Vert \varphi _{{\hat{\beta }}} - \varphi _\beta \Vert _{H^1(Q)} + \Vert \varphi _{{\hat{\beta }}} - \varphi _\beta \Vert _{C({\bar{Q}})} \le C_{10}\Vert {\hat{\beta }} - \beta \Vert _Z. \end{aligned}$$
(6.31)

Getting back to (6.26), and using (6.31), we get

$$\begin{aligned} \Vert u_{{\hat{\beta }}} - u_\beta \Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} \le C_{11}\Vert {\hat{\beta }} - \beta \Vert _Z. \end{aligned}$$
(6.32)

Using this in the equation satisfied by \(y_{{\hat{\beta }}} - y_\beta \) we also obtain

$$\begin{aligned} \Vert y_{{\hat{\beta }}} - y_\beta \Vert _{C({\bar{Q}})} \le C_{12}\Vert {\hat{\beta }} - \beta \Vert _Z. \end{aligned}$$
(6.33)

Now, (6.30)–(6.33) imply (6.18). Hence, we apply [10, Theorem 2.4] to deduce the existence of \({\bar{\varepsilon }} \in (0,\varepsilon ]\) and \({\tilde{\rho }} \in (0,{\bar{\rho }}]\) such that for every \(\gamma \in ({{\bar{\gamma }}}- {\bar{\varepsilon }},{{\bar{\gamma }}}+ {\bar{\varepsilon }})\) the system (6.12)–(6.15) has a solution \((y,\varphi ,u)\) with u in the interior of the ball \(B_{{\tilde{\rho }}}({\bar{u}})\) satisfying (6.16). Since \({\bar{\varepsilon }} \le \epsilon \) and \({\tilde{\rho }} \le {\bar{\rho }}\), we know that \(J_\gamma \) is strictly convex on \(B_{{\tilde{\rho }}}({\bar{u}})\), hence u is the unique local minimizer of \( \text{(Q }_\gamma ) \) in this ball. Moreover, \(u_\gamma = \gamma u\) belongs to the interior of the ball \(B_\rho (u_{{\bar{\gamma }}})\) and \(u_\gamma \) is the unique local minimizer of \( \text{(P }_\gamma ) \) in \(B_\rho (u_{{\bar{\gamma }}})\). Moreover, from (6.16) we infer

$$\begin{aligned} \Vert u_\gamma - u_{{\bar{\gamma }}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))}&\le \gamma \Vert u - {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} + |\gamma - {{\bar{\gamma }}}|\Vert {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))}\\&< ({{\bar{\gamma }}}+ {\bar{\varepsilon }})\lambda |\gamma - {{\bar{\gamma }}}| + |\gamma - {{\bar{\gamma }}}|\Vert {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))} = L|\gamma - {{\bar{\gamma }}}| \end{aligned}$$

for \(L = ({{\bar{\gamma }}}+ {\bar{\varepsilon }})\lambda + \Vert {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))}\). This ends the proof. \(\square \)