Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space

This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form ‖u(t)‖L1(Ω)≤γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u(t)\Vert _{L^1(\varOmega )} \le \gamma $$\end{document} for t∈(0,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in (0,T)$$\end{document}. This limits the total control that can be applied to the system at any instant of time. The L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-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 γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is investigated on the basis of different second order conditions.


Introduction
We study the optimal control problem (P) inf where κ > 0, We assume that Ω is a bounded, connected, and open subset of R n , n = 2 or 3, with a Lipschitz boundary , and that 0 < T < ∞ 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 (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 ∞ (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 ∞ (0, T ; L 2 (Ω)) 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 (Ω)) for all r > 4 4−n . Moreover, all these solutions belong to L ∞ (Q). This leads us to formulate the control problem in L ∞ (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 ∞ (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 ∞ . 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 (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 [1-5, 7-9, 11, 14-16, 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 u(t) 2 L 2 (Ω) ≤ 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 u(t) M(Ω) ≤ γ .
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 ∞ (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 γ is investigated.

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 ∈ L ∞ (Ω), a i j ∈ L ∞ (Ω) for every 1 ≤ i, j ≤ n, and for some Λ A > 0. We also assume that a : Q × R → R is a Carathéodory function of class C 2 with respect to the last variable satisfying the following properties: for almost all (x, t) ∈ Q.
Proof We decompose the state equation into two parts. First, we consider It is well known that it has a unique solution z ∈ W (0, T ) ∩ L ∞ (Q). Moreover, we have the estimates where C a is as in (2.2). Then, b(x, t, 0) = 0 and according to (2.2) ∂b ∂s (x, t, s) = ∂a ∂s (x, t, e |C a |t s + z(x, t)) + |C a | ≥ 0.
We consider the equation (2.11) Due to the properties of b, the existence and uniqueness of a solution w ∈ L ∞ (Q) ∩ W (0, T ) is well known; see [20,Theorem 5.5]. Moreover, the following estimates hold Denoting M = z L ∞ (Q) and using (2.4) we infer with the mean value theorem Combining this with (2.10) and (2.13) we get (2.14) for a non-decreasing function σ : [0, ∞) −→ [0, ∞).
To prove (2.7), we take φ = e −|C a |t y u and introduce the function f : Then, φ satisfies (2.16) Since f (x, t, 0) = 0 and ∂ f ∂s (x, t, s) ≥ 0, multiplying the above equation by φ, integrating in Ω, and using (2.1) we get Estimate (2.7) follows from this inequality as usual.

Theorem 2.2 The mapping G is of class C
Proof Let us consider the Banach space where X = Lr (0, T ; Lp(Ω)) + L r (0, T ; L 2 (Ω)), endowed with the norm Now, we define the mapping We have that F is of class C 2 , F(y u , y 0 , u) = (0, 0) for every u ∈ L r (0, T ; L 2 (Ω)), and is an isomorphism. Hence, an easy application of the implicit function theorem proves the result.
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 (Ω)) −→ R. From now on, we assume y d ∈ Lr (0, T ; Lp(Ω)), (2.19) wherer andp are defined in (2.3).

Corollary 2.1
If r > 4 4−n , then J is of class C 2 and its derivatives are given by the expressions Above A * denotes the adjoint operator of A The regularityφ ∈ C(Q) ∩ 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) −→ R and J (u) : L 2 (Q) × L 2 (Q) −→ R for every u ∈ L r (0, T ; L 2 (Ω)).

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 ∈ {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 (Ω) constraint implies that one looks for the action of a controlling laser whose optimal support is small; see [12].

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 callū a local minimizer for (P) in the L r (0, T ; L 2 (Ω)) sense with r > 4 4−n ifū ∈ U ad ∩ L ∞ (Q) and there exists ε > 0 such that It is immediate to check that ifū is a local minimizer in the L r (0, T ; L 2 (Ω)) sense, then it is also a local minimizer in the L r (0, T ; L 2 (Ω)) sense for every r < r ≤ ∞.

4)
ϕ + κū +μ = 0. (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 minimizerū, we takeȳ andφ as solutions of (3.2) and (3.3), respectively. Using the convexity of U ad and (2.20) we get Then, we can pass to the limit in the inequality J (ū)(u k −ū) ≥ 0 and, hence, we obtain This inequality is equivalent to the fact −(φ + κū) ∈ ∂ I U ad (ū) ⊂ L ∞ (Q). Here ∂ I U ad denotes the subdifferential of the indicator function I U ad : L 1 (Q) −→ [0, +∞], which takes the value I U ad (u) = 0 if u ∈ U ad and +∞ otherwise. Therefore, there existsμ ∈ ∂ I U ad such that (3.4) and (3.5) holds.

Let us denote by Proj
Then, we have the following consequence of the previous theorem.
Then, u ∈ 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 and for a.a. t ∈ (0, T ).
Since B γ ∩ L 2 (Ω) is a convex and closed subset of L 2 (Ω), the above inequality is the well known characterization of (3.7). Let us prove the first statement of (3.8). Take u(x, t) = sign(μ(x, t))|ū(x, t)|. Then, u ∈ U ad and with (3.4) we obtain which proves the desired identity. We prove the second statement of (3.8). For every ε > 0 we define Denote B ε the closed ball of L 1 (Ω) centered at 0 and radius ε. Take v ∈ B ε arbitrary. Then, we have that v +ū(t) ∈ B γ for t ∈ I ε , and (3.6) yields which implies thatμ(t) ≡ 0 in Ω for t ∈ I ε . Since ε > 0 is arbitrary, we infer the second statement of (3.8). Let us prove the third statement. Under the assumption For every ε > 0 and t ∈ (0, T ) we consider the sets We are going to prove that |Ω ε (t)| = 0 for almost all t ∈ (0, T ). Assume that |Ω ε (t)| > 0 for some ε > 0 and t ∈ (0, T ). Since |Ω ε (t)| > 0 by definition of the essential supremum, we can find two sets E ⊂ Ω ε (t) and F ⊂Ω ε (t) such that |E| = |F| > 0. We define the control 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

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 thatμ(t) ≡ 0 in Ω if and only if φ(t) L 1 (Ω) > κγ .

Corollary 3.3 Letū be as in Corollary 3.2. Then, we have the following propertȳ
This corollary is a straightforward consequence of (3.9).
To prove this theorem, we can argue as in the proof of Theorem 4.4 below to deduce the existence of K ∞ > 0 independent of γ such that ū L ∞ (Q) ≤ K ∞ . The last statement is a straightforward consequence of this estimate and the definition of γ 0 .

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 ∞ (Ω) and the non coercivity of J on this space. One can try to prove the existence of a solutionū of (P) in L 2 (Q) and then to deduce thatū ∈ L ∞ (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 : It can be easily checked that Theorem 4.1 For any M > 0 and all u ∈ L 2 (Q) the equation has a unique solution y M u ∈ 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 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 Let us define the mapping G M : L 2 (Q) −→ W (0, T ) associating to every u the corresponding solution y M u of (4.4).
This is a Banach space when it is endowed with the graph norm Now, we define the mapping Let us prove that the mapping First, we observe that a standard application of a Gagliardo-Nirenberg inequality leads to for every z ∈ W (0, T ). Using this inequality, (4.3), and the mean value theorem we infer .
Hence, F M is Fréchet differentiable. The continuity of D F M is immediate and, consequently, F M is of class C 1 . Using this and the continuity of the embedding ). An easy application of the implicit function theorem proves Theorem 4.2.
For every M > 0 we consider the control problems where y M u denotes the solution of (4.4). Problem (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 ∩ 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 ) ⊂ L 2 (Q).
From the chain rule and Theorem 4.2 we infer that J M : L 2 (Q) −→ R is of class C 1 and its derivative is given by the expression The proof of this theorem is the same as the one of Theorem 3.1. (4.14) Proof As in the proof for the first statement of (3.8), we have that (4.12) and (4.13) We denote by y 0 M 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 (P M ) and u ≡ 0 is an admissible control for (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 ϕ M by a constant independent of M. The idea of the proof is to make the substitution ϕ M ( where C a is given in (2.2). Then, ψ satisfies the equation which proves that u M is a solution of (P).

Remark 4.2
Let us compare problem (P) with the control problems where r ∈ ( 4 4−n , ∞). We observe that Theorems 2.1 and 2.2 , and Corollary 2.1 are applicable to deduce that any solution of (P r ) satisfies the optimality conditions (3.2)-(3.5). Then, the arguments of Theorem 4.4 apply to deduce that any solution of (P r ) belongs to L ∞ (Q). Let us check that problems (P) and (P r ) are equivalent in the sense that both have the same solutions. Indeed, since U ad ∩ L r (0, T ; L 2 (Ω)) ⊃ U ad ∩ L ∞ (Q), it is obvious that every solution of (P r ) is a solution of (P). Conversely, letū be a solution of (P) and take u ∈ U ad ∩ L r (0, T ; L 2 (Ω)) arbitrarily. For every integer k ≥ 1 we set u k = Proj [−k,+k] (u). Then, it is obvious that u k ∈ U ad ∩ L ∞ (Q) and u k → u in L r (0, T ; L 2 (Ω)). Using the optimality ofū we have J (ū) ≤ J (u k ) for all k, and passing to the limit we infer that J (ū) ≤ J (u). Since u was arbitrary, this implies thatū is a solution of (P r ).

Second Order Optimality Conditions
We consider the Lipschitz and convex mapping j : . 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 controlū ∈ U ad ∩ L ∞ (Q) satisfying the first order optimality conditions (3.2)-(3.5) we set We first prove the second order necessary conditions. Given an element v ∈ Cū, the classical approach to prove these second order conditions consists of taking a sequence {v k } ∞ k=1 converging to v such thatū + ρv k is a feasible control for (P) for every ρ > 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 nonlocal 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.
As a consequence, every element v of Cū satisfies (5.2).
Inserting this in the previous identity we obtain Sinceμv ≤ μ(t) L ∞ (Ω) |v|, we deduce from the above equality that J (ū)v = 0 if and only if (5.2) holds. Proof Let v be an element of Cū ∩ L ∞ (0, T ; L 2 (Ω)). We will prove that J (ū)v 2 ≥ 0. Later, we will remove the assumption v ∈ L ∞ (0, T ; L 2 (Ω)). Set For every integer k ≥ 1 we put where χ Ω 0 u(t) (x) takes the value 1 if x ∈ Ω 0 u(t) and 0 otherwise. Using that | Proj [−k,+k] (g(x, t))ū(x, t)| ≤ |v(x, t)| and the pointwise convergence Proj [−k,+k] (g(x, t))ū(x, t) → g(x, t)ū(x, t) almost everywhere in Q, we deduce with Lebesgue's Theorem that lim k→∞ a k (t) = a(t) for almost all t ∈ (0, T ). Therefore, we have that v k (x, t) → v(x, t) for almost all (x, t) ∈ Q. Moreover, we have Once again, with Lebesgue's Theorem we get v k → v in L r (0, T ; L 2 (Ω)) for every r < ∞. Let us prove that J (ū)v k = 0. To this end, we apply Lemma 5.1. Actually, we are going to prove that v k ∈ Cū. Given t ∈ I γ , taking into account (5.1) and the fact that where we used that v ∈ Cū in the last step.
In the case where ū(t) L 1 (Ω) < γ , 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 ∈ Cū, we deduce that v k also satisfies (5.2). Then, Lemma 5.1 implies that J (ū)v k = 0. Therefore, v k ∈ Cū holds.
Finally, we take v ∈ Cū arbitrary and for every k .
Proof We proceed by contradiction. If (5.3) is false for every δ > 0 and ε > 0, then for every integer k ≥ 1 there exists an element u k ∈ U ad such that Let us set ρ k = u k −ū L 2 (Q) and v k = (u k −ū)/ρ k . Then, we have v k L 2 (Q) = 1 and, taking a subsequence that we denote in the same way, we have v k v in L 2 (Q).
We divide the proof in several steps.
Step III -J (ū)v 2 ≤ 0. From (5.4) and a Taylor expansion we infer The strong convergenceū + θ k (u k −ū) →ū in L r (0, T ; L 2 (Ω)) yields the uniform convergences y θ k →ȳ and ϕ θ k →φ in L ∞ (Q), where y θ k and ϕ θ k are the state and adjoint state associated withū +θ k (u k −ū). This also implies that z θ k ,v k → z v strongly in L 2 (Q), where z v is the solution of (2.20) for y u =ȳ and z 2 θ k ,v k is the solution of (2.20) with v = v k and y u = y θ k . Then, we can pass to the limit in (5.7) when k → ∞ and deduce that J (ū)v 2 ≤ 0.
Step IV -Final contradiction. Since v ∈ Cū and J (ū)v 2 ≤ 0, according to the assumptions of the theorem, this is only possible if v = 0. Therefore, we have that v k 0 in L 2 (Q) and, consequently, z 2 θ k ,v k → 0 strongly in L 2 (Q). Now, using that v k L 2 (Q) = 1 and (2.21), we infer from (5.7) which contradicts our assumption κ > 0.
The next theorem establishes that the sufficient condition for local optimality, J (ū)v 2 > 0 for every v ∈ Cū \ {0}, provides a useful tool for the numerical analysis of the control problem. Given τ > 0 we define the extended cone
Proof First we prove the existence of τ > 0 and ν > 0 such that We proceed by contradiction. If (5.9) fails for all strictly positive numbers τ, ν, then for Dividing v k by its L 2 (Q) norm and taking a subsequence we get (5.11) We prove that v ∈ Cū. First, from (5.10) and (5.11) we get Thus, we have J (ū)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 (ū(t); v(t)) ≤ 0 for almost all t ∈ I γ . Now, from the identity J (ū)v = 0, (5.1), and (3.8) it follows μv dx dt.

This implies
μv dx dt. (5.12) Now we have From this identity and (5.12) we infer This inequality along with j (ū(t); v(t)) ≤ 0 for t ∈ I γ implies that j (ū(t); v(t)) = 0 for almost all t ∈ I + γ . We have proved that v ∈ Cū. From (5.10) we infer Sinceū satisfies the second order condition, the above inequality is only possible if v = 0. Therefore, we have that v k 0 in L 2 (Q). Using (2.21) and the fact that z v k → 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 ρ > 0 arbitrarily small, from Theorem 2.2 we deduce the existence of ε > 0 such that Using this estimate, we get from (2.17) and (2.22), and taking a smaller ε if necessary where z u,v = G (u)v, z v = G (ū)v, and ϕ u andφ are the adjoint states corresponding to u andū, respectively. Therefore, selecting ρ small enough we obtain with (2.21) for some ε > 0 Combining this with (5.9) we infer (5.8).

Stability of the Optimal Controls with Respect to
The aim of this section is to prove some stability of the local or global solutions of (P) with respect to γ . For every γ > 0 we consider the control problems First, we prove some continuity of the solutions of (P γ ) with respect to γ . Theorem 6.1 Let {γ k } ∞ k=1 ⊂ (0, ∞) be a sequence converging to some γ > 0. For every k let u γ k be a global minimizer of the problem (P γ k ). Then, the sequence {u γ k } ∞ k=1 is bounded in L ∞ (Q). Moreover, if u γ is a weak * limit in L ∞ (Q) of a subsequence of {u γ k } ∞ k=1 , then u γ is a global minimizer of (P γ ) and the convergence is strong in L p (Q) for every p < ∞. Reciprocally, for every strict local minimizer u γ of (P γ ) in the L r (0, T ; L 2 (Ω)) sense with 4 4−n < r < ∞, there exists a sequence {u γ k } ∞ k=1 such that u γ k is a L r (0, T ; L 2 (Ω)) local minimizer of (P γ k ) and u γ k → u γ strongly in L p (Q) for every p < ∞.

Proof
The boundedness of {u γ k } ∞ k=1 in L ∞ (Q) follows from Theorem 3.2. Therefore, we can take subsequences converging weakly * in L ∞ (Q). Let us take one of these subsequences, that we denote in the same form, such that u γ k * û in L ∞ (Q). Let u γ be a solution of (P γ ). For every k we define is a closed and convex subset of L 2 (Q) andû k û in L 2 (Q), we deduce thatû ∈ U γ . With the compactness of the embedding W (0, T ) ⊂ L 2 (Q) we can easily prove that yû k → yû in L 2 (Q). Using these convergences and the optimality of u γ k and u γ we get This implies that J (u γ ) = J (û) = lim k→∞ J (u γ k ). This identity proves thatû is a solution of (P γ ). Moreover, the convergence y u γ k → y u γ in L 2 (Q) leads to lim k→∞ u γ k L 2 (Q) = û L 2 (Q) . From this fact and the weak convergence u γ k u γ in L 2 (Q), we obtain that u γ k →û in L 2 (Q). This along with the boundedness of {u γ k } ∞ k=1 in L ∞ (Q) implies the strong convergence in L p (Q) for every p < ∞. Let us prove the second part of the theorem. Let u γ be an L r (0, T ; L 2 (Ω)) strict local minimizer to (P γ ). This means that there exists ε > 0 such that where B ε (u γ ) is the closed ball in L r (0, T ; L 2 (Ω)) of radius ε and center u γ . Now, we consider the problems It is immediate that u γ is the unique solution of (P B γ ). Observe that the controls u k defined in (6.1) are elements of U γ k ∩ B ε (u γ ) for all k large enough. Hence, U γ k ∩ B ε (u γ ) is non-empty, closed, convex, and bounded in L r (0, T ; L 2 (Ω)). Therefore, problem (P B γ k ) has at least one solution u γ k . Let us prove that u γ k → u γ in L p (Q) for every p < ∞. Denote y γ k and ϕ γ k the state and adjoint state associated with u γ k . Since {u γ k } ∞ k=1 is bounded in L r (0, T ; L 2 (Ω)) we infer from Theorem 2.1 the boundedness of {y γ k } ∞ k=1 in L ∞ (Q). Hence, from the adjoint state equation and the classical estimates for linear equations we deduce that {ϕ γ k } ∞ k=1 is also bounded in L ∞ (Q). Due to the optimality of u γ k for (P B γ k ) we obtain Setting S = U γ k ∩ B ε (u γ ) we get from the above inequalities where Proj S denotes the L 2 (Q) projection on S. Let us prove that For this purpose we define Then, it is obvious that The first two inequalities show that u ∈ S and, consequently, the third one contradicts the fact that u γ k is the L 2 (Q) projection of − 1 κ ϕ γ k unless |Q 0 | = 0. Now, the boundedness of {ϕ γ k } ∞ k=1 in L ∞ (Q) and (6.2) imply the boundedness of {u γ k } ∞ k=1 . Therefore, there exists a subsequence, denoted in the same way, such that u γ k * û in L ∞ (Q). Using the functions {û k } ∞ k=1 defined in (6.1) and arguing as above, we deduce thatû ∈ U γ . Moreover, is is also immediate thatû ∈ B ε (u γ ). Let us consider the functions {u k } ∞ k=1 defined in (6.1). Since we have that u k → u γ in L ∞ (Q) as k → ∞ and u k ∈ U γ k ∩ B ε (u γ ) for every k large enough. Then, using the optimality of u γ and u γ k , and the fact that u k andû are feasible controls for (P B γ k ) and (P B γ ), respectively, we infer This implies that J (u γ ) = J (û) and, hence,û is also a solution of (P B γ ). Due to the uniqueness of solution of (P B γ ) we conclude that u γ =û. The strong convergence u γ k → u γ in L p (Q) follows as above. We have proved that every subsequence converge to u γ , then the whole sequence does. In particular, the convergence u γ k → u γ in L r (0, T ; L 2 (Ω)) implies that u γ k is in the interior of the ball B ε (u γ ) for all k sufficiently large. Hence, u γ k is an L r (0, T ; L 2 (Ω)) local minimizer of (P B γ k ).

Remark 6.1
Given an L r (0, T ; L 2 (Ω)) strict local minimizer of (P γ ), from the above theorem we deduce the existence of a family {u γ } γ >0 of L r (0, T ; L 2 (Ω)) local minimizers of problems (P γ ) such that u γ → u γ in L p (Q) as γ → γ for every p < ∞. Looking at the definition of the elements u γ k in the previous proof we have that Theorem 6.2 Let {u γ } γ be a family of local minimizers of problems (P γ ) such that u γ → u γ in L r (0, T ; L 2 (Ω)) as γ → γ with u γ a local minimizer of (P γ ) 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 γ we definê Then we havê From here we infer that v γ ∈ U γ ∩ B ε (u γ ) for γ close enough to γ with B ε (u γ ) defined in (5.3). Therefore, we get In the case γ < γ , using (6.7), the optimality of u γ , and the definition ofv γ we obtain with the mean value theorem In the case γ > γ we proceed as follows From here we get which concludes the proof.
Theorems 5.2 and 6.2 imply Hölder stability with respect to γ of the optimal controls if the sufficient second order condition J (u γ )v 2 > 0 ∀v ∈ Cū \ {0} holds. Now, we are interested in proving Lipschitz stability. To this end we need to make a stronger assumption, namely whereγ > 0 is fixed and C 0 (Ω) denotes the space of continuous real valued functions onΩ vanishing on . From the first assumption in (6.8) we deduce the existence of strictly positive numbers ρ and ν such that and ∀u ∈ B ρ (uγ ), (6.9) where B ρ (uγ ) denotes the Lr (0, T ; L 2 (Ω)) closed ball. Indeed, if (6.9) does not hold, then we can take sequences 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 κ > 0. We finish this section by proving the next theorem.
Moreover, there exists a constant L such that Proof Let us take ρ > 0 such that (6.9) holds. Then, J has at most one local (and global) minimizer u γ in the closed set B ρ (uγ )∩U ad . This is a consequence of the strict convexity of J in the ball B ρ (uγ ); see (6.9). We will prove that this local minimizer belongs to the interior of the Lr (0, T ; L 2 (Ω)) ball B ρ (uγ ) if γ is close enough toγ , and consequently it is a local minimizer of (P γ ). In order to prove this, as well as (6.10), we reformulate the control problem (P γ ) as follows and y γ,u is the solution of the semilinear parabolic equation It is obvious that the problems (P γ ) and (Q γ ) are equivalent for every γ . This equivalence is understood in the sense that u is a local (global) minimizer of (Q γ ) if and only if u γ = γ u is a local (global) minimizer of (P γ ), and J (u γ ) = J γ (u); recall Remark 4.2.
Due to (6.9) and the fact that J γ (u)v 2 = γ 2 J (u γ )v 2 , we deduce that Therefore, J γ is strictly convex on the ball Bρ(ū). Hence, a control u is a local solution of (Q γ ) in the interior of Bρ(ū) if and only if u satisfies the optimality system ∂ y ∂t + Ay + a(x, t, y) = γ u in Q, y = 0 on Σ, y(0) = y 0 in Ω, 14) γ ϕ + κγ 2 u + μ = 0. (6.15) Denote byȳ andφ the state and adjoint state associated toū. Our goal is to apply [10,Theorem 2.4] to the previous optimality system. To this end we define the spaces: On Y and Φ we consider the graph norms .
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.