Abstract
This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form \(\Vert u(t)\Vert _{L^1(\varOmega )} \le \gamma \) for \(t \in (0,T)\). This limits the total control that can be applied to the system at any instant of time. The \(L^1\)-norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to \(\gamma \) is investigated on the basis of different second order conditions.
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1 Introduction
We study the optimal control problem
where \(\kappa > 0\),
with \(0< \gamma < +\infty \), and \(y_u\) is the solution of the semilinear parabolic equation
with
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
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:
for almost all \((x,t) \in Q\).
As usual W(0, T) denotes the Hilbert space
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
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
It is well known that it has a unique solution \(z \in W(0,T) \cap L^\infty (Q)\). Moreover, we have the estimates
see, for instance [17, Chapter III]. Now, we define \(b:Q \times {\mathbb {R}} \longrightarrow {\mathbb {R}}\) as follows
where \(C_a\) is as in (2.2). Then, \(b(x,t,0) = 0\) and according to (2.2)
We consider the equation
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
Denoting \(M = \Vert z\Vert _{L^\infty (Q)}\) and using (2.4) we infer with the mean value theorem
Combining this with (2.10) and (2.13) we get
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
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
Then, \(\phi \) satisfies
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
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
Proof
Let us consider the Banach space
where \(X = L^{{\hat{r}}}(0,T;L^{{\hat{p}}}(\varOmega )) + L^r(0,T;L^2(\varOmega ))\), endowed with the norm
Now, we define the mapping
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
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
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
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
Above \(A^*\) denotes the adjoint operator of A
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
where
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
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
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
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
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
and
Then, \(u \in {U_{ad}}\) and (3.4) yields
This is only possible if \(|I_v| = 0\). In order to prove (3.7) we use (3.5) and (3.6) to get
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
which proves the desired identity. We prove the second statement of (3.8). For every \(\varepsilon > 0\) we define
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
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
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
Since \(\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} = \gamma \), we get
Moreover, we get with the first statement of (3.8)
which contradicts (3.6) unless it is satisfied for a set of points t of zero Lebesgue measure. Taking
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
This yields
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
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.
-
(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).
-
(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.
-
(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
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(x, t, y) as follows. For every \(M > 0\) we define the function \(f_M:{\mathbb {R}} \longrightarrow {\mathbb {R}}\) by
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
for almost all \((x,t) \in Q\).
Theorem 4.1
For any \(M > 0\) and all \(u \in L^2(Q)\) the equation
has a unique solution \(y^M_u \in W(0,T)\). Moreover, \(y^M_u\) satisfies the inequalities
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
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
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
\(\square \)
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
where \(y^M_u = G_M(u)\).
Proof
Let us introduce the space
This is a Banach space when it is endowed with the graph norm
Now, we define the mapping
Let us prove that the mapping
is of class \(C^1\) with
First, we observe that a standard application of a Gagliardo–Nirenberg inequality leads to
for every \(z \in W(0,T)\). Using this inequality, (4.3), and the mean value theorem we infer
From here we deduce
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
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
where \(\varphi \in W(0,T)\) is the solution of the adjoint state equation
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
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
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
From this inequality we infer
Since \(u_M\) is solution of \( \text{(P }_M) \) and \(u \equiv 0\) is an admissible control for \( \text{(P }_M) \) we get
This leads to
Using again (4.5) and this estimate we deduce
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
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
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)
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
which proves that \(u_M\) is a solution of (P). \(\square \)
Remark 4.2
Let us compare problem (P) with the control problems
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
where
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
where
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
As a consequence, every element v of \(C_{{\bar{u}}}\) satisfies (5.2).
Proof
From (2.20), (3.5), and (3.8) we infer
Using that \(j'({\bar{u}}(t);v(t)) = 0\) for almost all \(t \in I_\gamma ^+\) and (5.1) we get
Inserting this in the previous identity we obtain
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
From (5.1) we infer
For every integer \(k \ge 1\) we put
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
and
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
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(x, t). 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
Then, we have for each fixed k and \(\forall \rho \in (0,\rho _k)\)
Using this estimate we have that \(\Vert {\bar{u}}(t) + \rho v_k(t)\Vert \le \gamma \) if \(j({\bar{u}}(t)) = \gamma \) and \(0< \rho < \rho _k\):
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
If \(\Vert {\bar{u}}(t)\Vert _{L^1(\varOmega )} < \gamma - \frac{1}{k}\), then we get
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
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
Then, we have
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
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
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 \(\theta _k \in (0,1)\)
Dividing this inequality by \(\rho _k\) we obtain
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 \(t \in I_\gamma \)
Take a measurable subset \(J \subset I_\gamma \). Since the functional
is continuous and convex, recall (5.1), the weak convergence \(v_k \rightharpoonup v\) in \(L^2(Q)\) implies
Since \(J \subset I_\gamma \) is an arbitrary measurable set, we infer for almost all \(t \in I_\gamma \)
Identities (3.5) and \(J'({\bar{u}})v = 0\), and (3.8) imply
From (5.5) we deduce
The last two relations lead to
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
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
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
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)
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
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
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
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
We prove that \(v \in C_{{\bar{u}}}\). First, from (5.10) and (5.11) we get
Thus, we have \(J'({\bar{u}})v = 0\). Let us set
Then, we obtain with (5.10) and (5.11)
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
This implies
Now we have
From this identity and (5.12) we infer
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
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
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
Using this estimate, we get from (2.17) and (2.22), and taking a smaller \(\varepsilon \) if necessary
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 \(\varepsilon > 0\)
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
where
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
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
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
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
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
Setting \(S = {U_{\gamma _k}}\cap B_\varepsilon (u_\gamma )\) we get from the above inequalities
where \({\text {Proj}}_S\) denotes the \(L^2(Q)\) projection on S. Let us prove that
For this purpose we define
Put
Then, it is obvious that
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
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
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
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
Proof
The first part of the theorem follows from Remark 6.1. We only have to prove (6.4). For every \(\gamma '\) we define
Then we have
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
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
In the case \(\gamma ' > \gamma \) we proceed as follows
From here we get
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
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
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
It is easy to pass to the limit and to deduce
This inequality and (6.8) yield \(v = 0\). But, arguing as in the proof of Theorem 5.3 we infer
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
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
where
and \(y_{\gamma ,u}\) is the solution of the semilinear parabolic equation
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 \(u_\gamma = \gamma u\)
Due to (6.9) and the fact that \(J_\gamma ''(u)v^2 = \gamma ^2 J''(u_\gamma )v^2\), we deduce that
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
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:
On \({\mathcal {Y}}\) and \(\varPhi \) we consider the graph norms
Thus, X is a Banach space. Moreover, we introduce the mapping \(f:X \times Y \longrightarrow Z\) and the multivalued function \(F:X \longrightarrow Z\)
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
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
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
for some \(\lambda > 0\). For this purpose it is enough to prove that the equation
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
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
where
and y satisfies the equation
Let us consider the solution \(\xi _\beta \in {\mathcal {Y}}\) of the equation
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
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
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
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
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
Given \(\beta , {\hat{\beta }} \in Z\), we infer from (6.23)-(6.24) and (6.25) for \(\beta \) and \({\hat{\beta }}\)
Adding these inequalities we get
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
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
Now, from the equations satisfied by \(y_{{\hat{\beta }}} - y_\beta \) and \(\varphi _{{\hat{\beta }}} - \varphi _\beta \) we get
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
Combining this inequality with (6.26) and using (6.9) we deduce
This yields
Using (6.27) and (6.29) it follows that
Now, (6.28) and (6.30) lead to
Getting back to (6.26), and using (6.31), we get
Using this in the equation satisfied by \(y_{{\hat{\beta }}} - y_\beta \) we also obtain
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
for \(L = ({{\bar{\gamma }}}+ {\bar{\varepsilon }})\lambda + \Vert {\bar{u}}\Vert _{L^{{\hat{r}}}(0,T;L^2(\varOmega ))}\). This ends the proof. \(\square \)
References
Casas, E., Chrysafinos, K.: Analysis of the velocity tracking control problem for the 3D evolutionary Navier–Stokes equations. SIAM J. Control. Optim. 54(1), 99–128 (2016)
Casas, E., Clason, C., Kunisch, K.: Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control. Optim. 51(1), 28–63 (2013)
Casas, E., Herzog, R., Wachsmuth, G.: Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations. ESAIM Control Optim. Calc. Var. 23, 263–295 (2017)
Casas, E., Kunisch, K.: Parabolic control problems in space-time measure spaces. ESAIM Control Optim. Calc. Var. 22(2), 355–370 (2016)
Casas, E., Kunisch, K.: Stabilization by sparse controls for a class of semilinear parabolic equations. SIAM J. Control. Optim. 55(1), 512–532 (2017)
Casas, E., Kunisch, K.: Optimal control of the 2d evolutionary Navier–Stokes equations with measure valued controls. SIAM J. Control. Optim. 59(3), 2223–2246 (2021)
Casas, E., Mateos, M., Rösch, A.: Approximation of sparse parabolic control problems. Math. Control Relat. Fields 7(3), 393–417 (2017)
Casas, E., Ryll, C., Tröltzsch, F.: Second order and stability analysis for optimal sparse control of the FitzHugh–Nagumo equation. SIAM J. Control. Optim. 53(4), 2168–2202 (2015)
Casas, E., Vexler, B., Zuazua, E.: Sparse initial data identification for parabolic PDE and its finite element approximations. Math. Control Relat. Fields 5(3), 377–399 (2015)
Dontchev, A.L.: Implicit function theorems for generalized equations. Math. Program. 70, 91–106 (1995)
Garcke, H., Lam, K.F., Signori, A.: Sparse optimal control of a phase field tumor model with mechanical effects. SIAM J. Control. Optim. 52(2), 1555–1580 (2021)
Gugat, M., Mateos, M., Tröltzsch, F.: Exponential stability for the Schlögl system by Pyragas feedback. Vietnam J. Math. 48, 769–790 (2020)
Gunzburger, M., Manservisi, S.: The velocity tracking problem for Navier–Stokes flows with bounded distributed control. SIAM J. Control. Optim. 37(6), 1913–1945 (1999)
Herzog, R., Stadler, G., Wachsmuth, G.: Directional sparsity in optimal control of partial differential equations. SIAM J. Control. Optim. 50(2), 943–963 (2012)
Kunisch, K., Pieper, K., Vexler, B.: Measure valued directional sparsity for parabolic optimal control problems. SIAM J. Control. Optim. 52(5), 3078–3108 (2014)
Kunisch, K., Trautmann, P., Vexler, B.: Optimal control of the undamped linear wave equation with measure valued controls. SIAM J. Control. Optim. 54(3), 1212–1244 (2016)
Ladyzhenskaya, O., Solonnikov, V., Ural’tseva, N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1988)
Leykekhman, D., Vexler, B., Walter, D.: Numerical analysis of sparse initial data identification for parabolic problems. ESAIM Math. Model. Numer. Anal. 54(4), 1139–1180 (2020)
Robinson, S.M.: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16(2), 292–309 (1991)
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Philadelphia (2010)
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Eduardo Casas was supported by MCIN/ AEI/10.13039/501100011033/ under research projects MTM2017-83185-P and PID2020-114837GB-I00. Karl Kunisch was supported by the ERC advanced Grant 668998 (OCLOC) under the EU’s H2020 research program.
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Casas, E., Kunisch, K. Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space. Appl Math Optim 85, 12 (2022). https://doi.org/10.1007/s00245-022-09850-7
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DOI: https://doi.org/10.1007/s00245-022-09850-7
Keywords
- Optimal control
- Semilinear parabolic equations
- Non-smooth control constraints
- First and second order optimality conditions