Infinite Horizon Optimal Control for a General Class of Semilinear Parabolic Equations

A class of infinite horizon optimal control problems subject to semilinear parabolic equations is investigated. First and second order optimality conditions are obtained, in the presence of constraints on the controls, which can be either pointwise in space-time, or pointwise in time and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} in space. These results rely on a new L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} estimate for nonlinear parabolic equations in an essential manner.

By investigating (P) we continue our efforts on studying infinite horizon optimal control problems with semilinear parabolic equations as constraints. In [8] the nonlinearities were chosen of polynomial type, no constraints were enforced on the controls, and the focus was put on nonsmooth, sparsity enhancing control costs, which entail that the controls settle down at zero once the states enter into a neighborhood of a stable equilibrium. Later, in [9] the nonlinearity was not restricted to be a polynomial and the conditions on f were very similar to those imposed in the present paper. The same type of control constraints were imposed as well. The major step forward in the current paper compared to [9] consists in an L ∞ (Q) estimate of the states for feasible controls, i.e. for controls with the property that the associated states y u are in L 2 (Q). Utilizing this property, well-posedness and C 2 regularity of the control-tostate mapping, associating the infinite horizon controls to the infinite horizon states, can be guaranteed, and a second order analysis of (P) becomes possible. This was not the case in [9], where the first order conditions of the infinite horizon problem were obtained as the limit of the associated finite horizon problems, and no second order analysis was carried out. The authors are not aware about the availability of the second order analysis for optimal control problems with constraints as in (1.2) even in the finite horizon case. Along a related, but different line of research we also investigated infinite horizon optimal control problems with a discount factor on the state, [11] and [12]. This allows to treat a larger class of nonlinearities at the expense of less information of the optimal states as time increases.
Most of the literature on infinite horizon problems is carried out for ordinary differential equations. Let us mention some of these contributions. In [7] the importance of infinite horizon problems in applications is stressed. In general, when formulating optimal control problems, the time horizon can be subject to ambiguity. In such cases the choice as infinite horizon problem can be a valuable choice. The first article, focusing on infinite horizon problems may be [15]. More recent contributions all in the context of ordinary differential equations are available for instance in [1,2,4]. Concerning the literature, pointwise constraints as in (1.3) have received considerably more attention than norm constraints as in (1.2). However, from a practical point of view (1.2) appears to be equally important. In the case of optimal control of Navier-Stokes equations the suitability of this type of constraints was discussed in [14]. The use of the L 1 (ω) norm replacing the L 2 (ω) was studied in [10]. The last two references were devoted to final horizon control problems.
Briefly, the paper is structured in the following way. In Sect. 2, the existence of optimal controls and first order optimality conditions are established. Necessary and sufficient second order conditions for the two choices of K ad in (1.2) and (1.3) are obtained in Sect. 3. Section 4 is devoted to convergence results for the finite horizon problems associated to (P), to the infinite horizon problem. This is not only of intrinsic interest but also of relevance for numerical realization. In the Appendix the relevant results for the state equation, and the associated linearized and adjoint equations are established. The L ∞ (Q) regularity result for the state equation, already mentioned above, may be of interest beyond its application in optimal control.

Existence of an optimal control and first order optimality conditions
In this section, we prove the existence of an optimal solution of (P) and derive the first order optimality conditions satisfied by any local minimizer. For this purpose we will also address the issue of differentiability of the relation control-to-state and of the cost functional J . Let us observe that Theorem A.2 implies the existence of a unique state y u for every control u ∈ U ad . However, it could happen that y u / ∈ L 2 (Q) and, consequently, J (u) = ∞. Therefore, the assumption about the existence of a control u 0 ∈ U ad such that J (u 0 ) < ∞ is needed. This issue will not be addressed in this paper, the reader is referred, for instance, to [3] and [8] for this question. We will say that u is a feasible control if u ∈ U ad and J (u) < ∞.
as norm. It is well known that (W (0, T ), · W (0,T ) ) is a Banach space. In fact, it is a Hilbert space because · W (0,T ) is a Hilbertian norm. Furthermore, the embedding and {y u k } ∞ k=1 in L 2 (Q ω ) and L 2 (Q), respectively, follows. Then, taking subsequences we can assume that (u k , y u k ) (ū,ȳ) in L 2 (Q ω ) × L 2 (Q). Since U ad is a closed and convex subset of L 2 (Q ω ), we infer thatū ∈ U ad . Due to the weak lower semicontinuity of J with respect to (y, u) in L 2 (Q)×L 2 (Q ω ), it is enough to establish thatȳ is the state associated toū to conclude the proof. For this purpose we have to show thatȳ satisfies (A.2) with g + χ ωū on the right hand side for every T < ∞. The only delicate point in this respect is to prove the convergence of f (x, t, Hence, using the compactness of the embedding W (0, T ) ⊂ L 2 (Q T ) the desired convergence follows.
Hereafter, the following additional hypothesis on f is assumed: Let us denote for every p satisfying (1.10) and by G p : U p −→ Y the mapping G p (u) = y u , where y u is the solution of (1.1). Y p is a Banach space when endowed with the associated graph norm. We observe that U ∞ ⊂ U p and G ∞ is the restriction of G p to U ∞ .

Theorem 2.2 Let us assume that U p is not empty. Then, U p is an open subset of
Proof The proof will be based on the implicit function theorem. For this purpose we define the mapping By definition of Y p and using (1.8), we deduce that F p is well defined and is of class C 1 . Further, we have that F p (y u , u) = (0, 0) for every u ∈ U p and Then, with continuous dependence. This is an immediate consequence of Theorem A.3 with d(x, t, s) = ∂ f ∂ y (x, t, s) and y = y u ∈ L ∞ (Q). Finally, the theorem follows by applying the implicit function theorem.
As a consequence of the above theorem, we have that J : U p −→ R is well defined. The next theorem establishes its differentiability.

Theorem 2.3
Assuming that U p is not empty, the functional J is of class C 1 and for every u ∈ U p and v ∈ L 2 (Q ω ) ∩ L p (0, ∞; L 2 (ω)) its derivative is given by The fact that J is of class C 1 is an immediate consequence of Theorem 2.2 and the chain rule. Formula (2.4) is deduced in the standard way from Eqs. (2.2) and (2.5). Concerning the well posedness of (2.5) we refer to Theorem A.4.
We conclude this section establishing the first order optimality conditions satisfied by every local minimizer of (P) and deducing some consequences from them. In this paper, a local minimizerū is understood in the L 2 (Q ω ) sense and it is assumed that u ∈ U ∞ ∩ U ad .

Theorem 2.4 Letū be a local minimizer of (P). Then
This theorem is an immediate consequence of Theorem 2.3 and the inequality J (ū)(u −ū) ≥ 0 for all u ∈ U ad . Corollary 2.1 Letφ andū satisfy (2.7) and (2.8). If K ad is given by (1.2), then the following properties hold for almost all t ∈ (0, ∞) In the case that K ad is given by (1.3), then we havē In both cases we have thatū ∈ L ∞ (Q ω ).

Second order optimality conditions
In this section we address the second order optimality conditions for (P). For this purpose, in addition to assumptions (1.4)-(1.7) we impose the following hypotheses: f : Q × R −→ R is of class C 2 with respect to the second variable and satisfies for almost all (x, t) ∈ Q. We observe that (3.1) and (3.2) imply (2.1). Indeed, it is enough to select Then, using the mean value theorem we infer for almost all ( 2) and supposing that U p is not empty, G p : The C 2 differentiability of G follows from the implicit function theorem applied to the mapping F p introduced in the proof of Theorem 2.2. It is enough to observe that now F p is of class C 2 . The Eq. (3.4) follows differentiating the identity F p (G p (u), u) = 0 twice.
As a consequence of Theorem 3.1 and the chain rule we have the following corollary.
Corollary 3.1 If U p is not empty, then the function J : U p −→ R is of class C 2 and we have for every u ∈ U and v 1 , v 2 ∈ L 2 (Q ω ) ∩ L p (0, ∞; L 2 (ω)). The analysis of second order optimality conditions is carried out in the next two subsections, where we consider the cases with K ad given by (1.2) or (1.3).

Case I: K ad
For this case we consider the Lagrange function Theorem 2.3 and Corollary 3.1 imply that L is of class C 2 and we have the expressions (3.7) The identities (3.6) and (3.7) define continuous linear and bilinear forms on L 2 (Q ω ) and L 2 (Q ω ) 2 , respectively.
Proof Using (3.6), (2.10), (2.11), and (2.15) we infer In order to formulate the second order optimality conditions we introduce the cone of critical directions associated withū: Then we have the following second order necessary optimality conditions.

Theorem 3.2 Ifū is a local minimizer of (P), then
Proof Sinceū is a local minimizer of (P), there exists ε > 0 such that The assumption v ∈ L ∞ (0, ∞; L 2 (ω)) will be removed later. Let us fix an integer Hence, the mapping φ k is well defined and it is of class C ∞ . Let us prove some properties of this mapping.
Then, using that v ∈ Cū we deduce that the last integral in the above inequality is less than or equal to zero and, consequently, From the definition of α k and k ≥ k 0 > 1 γ 2 we obtain .
holds. Then, we have Arguing as in the previous step and using again the definition of α k and k 0 with . From the local optimality ofū and the established properties of φ k we infer that Using (2.10), (2.11), (2.15), and (2.16) we obtain that Inserting this in the above inequality we infer with (3.7) Therefore from (2.10) and the above relations we deduce Now, we give a second order sufficient optimality condition. Theorem 3.3 Letū ∈ U ad ∩U ∞ satisfy the first order optimality conditions (2.6)-(2.8) and the second order condition ∂ 2 L ∂u 2 (ū, μ)v 2 > 0 for every v ∈ Cū\{0}. Then, there exists κ > 0 and ε > 0 such that Proof We argue by contradiction and assume that (3.9) does not hold. Then, for every integer k ≥ 1 there exists a control u k ∈ U ad such that Then, The rest of the proof is split into three steps. Step (3.11) Using the differentiability of the mapping J : U p −→ R we infer with the mean value theorem and (3.10) , and ϕ θ k is the adjoint state corresponding to Then, it is straightforward to pass to the limit in the above expression and to get J (ū)v ≤ 0. This inequality and (3.11) imply that J (ū)v = 0. Next, taking into account that u k (t) L 2 (ω) ≤ γ for almost all t > 0, we have for Then, from the convergence v k v in L 2 (Q ω ) and the fact that φū ∈ L 2 (Q ω ) we infer from the above inequality This is possible if and only if ωū (t)v(t) dx ≤ 0 for almost all t ∈ I γ . Finally, we prove that this integral is 0 if t ∈ I + γ . For this purpose we use Lemma 3.1, (3.6), and the fact that J (ū)v = 0 as follows This inequality and (3.10) imply Performing a Taylor expansion and using again Lemma 3.1 we infer for some ϑ k ∈ [0, 1] Dividing the above inequality by Denoting by u ϑ k =ū + ϑ k (u k −ū), y ϑ k its associated state, and ϕ ϑ k the corresponding adjoint state, we get from (3.7) (3.13) where z ϑ k ,v k satisfies the equation (3.14) Now, we study the lower limit of (3.12). From Theorem A.3 and the boundedness of {v k } ∞ k=1 and {y ϑ k } ∞ k=1 in L 2 (Q ω ) and L ∞ (Q), respectively, we infer the boundedness W (0, ∞). Therefore, we can extract a subsequence, that we denote in the same way, such that Using this and the convergence v k v in L 2 (Q ω ), it is straightforward to pass to the limit in (3.14) and to deduce that z ϑ k ,v k z v in W (0, ∞), where z v is the solution of (2.2). Further, the convergence of y ϑ k →ȳ in Y p implies the convergence in L p (0, ∞; L 2 (Ω)) ∩ L ∞ (Q). Then, from Theorem A.4 we infer that ϕ ϑ k →φ in W (0, ∞) ∩ L ∞ (Q). Indeed, subtracting the equations satisfied by ϕ ϑ k andφ we get Then, using (3.3), the established convergence y ϑ k →ȳ, (A.21), and (A.22) we get the claimed convergence of {ϕ ϑ k } ∞ k=1 toφ. Now, we take the lower limit in (3.12). For this purpose we take into account that ∞). Hence, we get by (3.12) Below we prove that Thus, (3.7) and (3.15)-(3.16) yield ∂ 2 L ∂u 2 (ū, λ)v 2 ≤ 0. Let us prove (3.16). Given ε > 0, (2.7) implies the existence of T ε > 0 such that implies that I 2 → 0 as well. For I 3 we have where we have used again the boundedness of Since ε > 0 is arbitrarily small, we deduce the convergence I 3 → 0 as k → ∞.
Step III-Final contradiction The facts proved in Steps I and II along with the assumption ∂ 2 L ∂u 2 (ū,λ)v 2 > 0 for every v ∈ Cū\{0} lead to v = 0 and z v = 0. Therefore, looking at the relations (3.15) we obtain with (3.16) and v k L 2 (Q ω ) = 1 which contradicts the assumption ν > 0.

Case II: K ad = {v ∈ L 2 (!) :˛≤ v(x) ≤ˇfor a.a. x ∈ !}.
In this case, the cone of critical directions is defined by Analogously to Theorem 3.2 we have the following result.
Proof Sinceū is a local minimizer of (P), there exists ε > 0 such that }, thenū + ρv k ∈ U ad ∩ B ε (ū) for every ρ ∈ (0, ρ k ). In view of (2.13), it is straightforward to check that the condition J (ū)v = 0 in the definition of Cū is equivalent to (φ + νū)(x, t)v(x, t) = 0 for almost all (x, t) ∈ Q ω . Using this fact, it is immediate that J (ū)v k = 0 for every k. Then, performing a Taylor expansion we get for every ρ ∈ (0, ρ k ) . Hence, we can pass to the limit in the previous inequality and obtain J (ū)v 2 ≥ 0. Now, we establish the sufficient second order conditions for local optimality.
Theorem 3.5 Letū ∈ U ad ∩U ∞ satisfy the first order optimality conditions (2.6)-(2.7) and the second order condition J (ū)v 2 > 0 for every v ∈ Cū\{0}. Then, there exists κ > 0 and ε > 0 such that The proof of this theorem follows by contradiction similarly to the proof of Theorem 3.3 with the obvious simplifications due to the constraints under consideration in this second case for U ad . For the proof of these results for finite horizon control problems the reader is also referred to [5,13]. The difficulties resulting from the infinite horizon can be overcome by following the arguments used in the proof of Theorem 3.3.

Approximation by finite horizon problems
In this section we consider the approximation of (P) by finite horizon optimal control problems and provide error estimates for these approximations. For every 0 < T < ∞ we consider the control problem  For every control u ∈ L 2 (Q T ,ω ) with associated state y T ,u and adjoint state ϕ T ,u we define extensions to Q ω and Q, denoted byû,ŷ T ,u , andφ T ,u , by setting (û,φ T ,u )(x, t) = (0, 0) if t > T andŷ T ,u is the solution of (1.1) associated with the extensionû. In this section, we assume that 0 ∈ K ad . Hence, if u ∈ U T ,ad , then u ∈ U ad holds. Given a local minimizer u T of (P T ), we denote by y T and ϕ T its associated state and adjoint state, respectively. Then, (u T , y T , ϕ T ) satisfies the optimality conditions established in Theorem 2.4 with Q and Q ω replaced by Q T and Q T ,ω . As a consequence, Corollary 2.1 is also satisfied by (u T , y T , ϕ T ) with the same changes.
In case U ad is given by (1.2), we define λ T (t) = ϕ T (t) + νu T (t) L 2 (ω) for t ∈ (0, T ) and the Lagrange function for every p ∈ ( 4 4−n , ∞]. Arguing as in Lemma 3.1 we also have The next two theorems establish the convergence of the approximating problems (P T )to (P) as T → ∞. . Every weak limitū in L 2 (Q ω ) of a sequence {û T k } ∞ k=1 with T k → ∞ as k → ∞ is a solution of (P). Moreover, strong convergenceû T k →ū in L p (0, ∞; L 2 (ω)) holds for every p ∈ [2, ∞).
Proof Since U T ,ad is not empty, the existence of solution for (P T )is a classical result. Actually, one can easily adapt the existence proof of solution for (P) to (P T ). We denote byỹ T the extension of y T by zero in Ω × (T , ∞). We point out thatỹ T =ŷ T . Let y 0 be the solution of (1.1) corresponding to u 0 . By definition of feasible control we have that J (u 0 ) < ∞. Using the optimality of u T we obtain This proves the boundedness of {û T } T >0 and {ỹ T } T >0 in L 2 (Q ω ) and L 2 (Q), respectively. Let {(û T k ,ỹ T k )} ∞ k=1 be a sequence with T k → ∞ as k → ∞ converging weakly to (ū,ȳ) in L 2 (Q ω )× L 2 (Q). Since {û T k } ∞ k=1 ⊂ U ad and U ad is closed in L 2 (Q ω ) and convex, we infer thatū ∈ U ad . Moreover, we can apply Theorem A.2 to the Eq. (4.1) and deduce the existence of a constant M 1 independent of k such that for all k ≥ 1 The two above estimates and (4.1) imply that for a constant independent of k. Using the convergence ofỹ k ȳ in L 2 (Q), the compactness of the embedding W (0, T ) ⊂ L 2 (Q T ) for every T < ∞, and the above estimate, it is obvious to pass to the limit in the equation for each T k ≥ T , and to deduce thatȳ is the solution of (4.1) associated toū for arbitrary 0 < T < ∞. This proves thatȳ is the solution of (1.1) corresponding toū. Further, sinceȳ ∈ L 2 (Q), we deduce thatū ∈ U ∞ . Let us prove thatū is a solution of (P). For every feasible control u of (P) we have This proves thatū is a solution of (P). Moreover, replacing u byū in the above inequalities we infer This convergence along with the weak convergence (û T k ,ỹ T k ) (ū,ȳ) in L 2 (Q ω ) × L 2 (Q) implies the strong convergence. Finally, for any p ∈ (2, ∞) we have Theorem 4.2 Letū be a strict local minimizer of (P). Then, there exist T 0 ∈ (0, ∞) and a family {u T } T >T 0 of local minimizers to (P T )such that the convergenceû T →ū in L p (0, ∞; L 2 (ω)) holds as T → ∞ for every p ∈ [2, ∞).
Proof Sinceū is a strict local minimizer of (P), there exists ρ > 0 such that J (ū) < J (u) for every u ∈ U ad ∩ B ρ (ū) with u =ū, where B ρ (ū) is the closed ball in L 2 (Q ω ) centered atū and radius ρ > 0. We consider the control problems Obviouslyū is the unique solution of (P ρ ). Existence of a solution u T of (P T ,ρ )is straightforward. Then, arguing as in the proof of Theorem 4.1 and using the uniqueness of the solution of (P ρ ), we deduce the convergenceû T →ū in L 2 (Q ω ) as T → ∞. This implies the existence Hence, u T is also a local minimizer of (P T )for T > T 0 . The strong convergencê u T →ū in L p (0, ∞; L 2 (ω)) follows from the convergence in L 2 (Q ω ) and the fact that û T L ∞ (0,∞;L 2 (ω)) ≤ γ for every T > 0.
In the previous theorem we proved the existence of local minimizers {u T } T >T 0 of problems (P T )converging toū assuming thatū is a strict local minimizer of (P). Moreover, in the proof of the theorem, the existence of an L 2 (Q ω )-closed ball B ρ (ū) such that the minimum of J T on the set U ad ∩ B ρ (ū) is achieved at the local minimizer u T was established. In particular, this implies that J T (u T ) ≤ J T (ū) for every T > T 0 . In the next theorem the following question is addressed: if {u T } T >T 0 is a sequence of local minimizers of problems (P T )converging toū, does the inequality J T (u T ) ≤ J T (ū) hold for T large enough? The positive answer to this question is also important to establish the estimates in Theorem 4.4 below.

Theorem 4.3 Suppose that U ad is defined by (1.2) or (1.3). Letū be a local minimizer of (P) satisfying the second order sufficient optimality condition given in Theorems 3.3 and 3.5, respectively. Let {u T } T >T 0 be a sequence of local minimizers of problems
The proof is carried out under the assumption that U ad is given by (1.2). It is similar, even easier, if U ad is given by (1.3). First, we observe that the convergencê u T →ū in L 2 (Q ω ) and the fact that û T (t) L 2 (ω) ≤ γ for almost every t > 0 implies thatû T →ū strongly in L p (0, ∞; L 2 (ω)) for every p < ∞. Then, for fixed p > 4 4−n , there existsT ≥ T 0 such thatû T ∈ U p for every T ≥T . This yieldŝ y T = G p (û T ) → G p (ū) =ȳ in Y p as T → ∞. Given the adjoint state ϕ T associated with u T , we denote byφ T its extension by 0 for t > T .
We proceed by contradiction. If the statement fails, then there exists a sequence . Taking a subsequence, denoted in the same way, we have v T k v in L 2 (Q ω ). Now, we split the proof in three steps.
Step Iφ T →φ in W (0, ∞) ∩ L ∞ (Q) as T → ∞. Let us set ψ T =φ T −φ and denote by χ T the real function taking the value 1 if t ∈ [0, T ] and 0 otherwise. Then, ψ T satisfies the equation

. Hence, with the mean value theorem and (3.2) we obtain that
). Moreover, from Theorem A.4 and the fact that y d ∈ L 2 (Q)∩L p (0, ∞; L 2 (Ω)), we get thatφ T is bounded in W (0, ∞)∩L ∞ (Q). Therefore the first term of the right hand side in the above partial differential equation converges to 0 in L q (0, ∞; L 2 (Ω)). The same convergence is true for the second term Step II v ∈ Cū. Using the local optimality ofū we get On the other side, using the convergence established in Step I and the convergencê u T k →ū in L 2 (Q ω ) along with the local optimality of u T k we infer The last two inequalities imply that J (ū)v = 0. Now, the proof continues as in the Step I of the proof of Theorem 3.3.
Step III-Contradiction Since (u T k , ϕ T k ) satisfies (2.10), we deduce the inequality L T k (ū, λ T k ) < L T k (u T k , λ T k ) with (4.3) and the fact that . Hence, performing a Taylor expansion and using (4.2) we infer Dividing the above expression by ρ 2 k /2 we get We observe that for k → ∞ Setting u θ k =ū + θ k (u T k −ū), we denote by y θ k the solution of (4.1) corresponding to the control u θ k and by ϕ θ k the corresponding adjoint state in Q T k ,ω . Then, putting Arguing as in Step I we obtain that ψ k → 0 in W (0, ∞) ∩ L ∞ (Q). Then, arguing as in Steps II and III of the proof of Theorem 3.3 and using the established convergences, we infer that ∂ 2 L ∂u 2 (ū,λ)v 2 ≤ 0 and the contradiction follows. Under an extra assumption on f , the following theorem provides estimates for the differenceû T −ū. (1.2) or (1.3) and thatū is a local minimizer of (P) satisfying the second order sufficient optimality condition. We assume that ∂ f ∂ y (x, t, y) ≥ 0 holds for all y ∈ R and almost all (x, t) ∈ Q. Let {u T } T >T 0 be a sequence of local minimizers of problems (P T )such thatû T →ū in L 2 (Q ω ). Then, there exist T * ∈ [T 0 , ∞) and a constant C such that for every T ≥ T * û T −ū L 2 (Q ω ) + ŷ T −ȳ W (0,∞) ≤ C y T (T ) L 2 (Ω) + y d L 2 (T ,∞;L 2 (Ω)) + g L 2 (T ,∞;L 2 (Ω)) .

Theorem 4.4 Suppose that U ad is defined by
(

4.4)
Proof We use the inequalities (3.9) or (3.17). For this purpose, we take T * ∈ [T * 0 , ∞) such that û T −ū L 2 (Q ω ) < ε for all T ≥ T * , where T * 0 is introduced in Theorem 4.3. Then, given T ≥ T * , (3.9) or (3.17), and Theorem 4.3 yield which leads to To prove the first estimate of (4.4) we observe thatŷ T satisfies the equation Testing this equation withŷ T , and using that f (x, t,ŷ T )ŷ T ≥ 0 due to the monotonicity of f with respect to y and (1.4), it follows that From this inequality we infer with (1.9) that This inequality and (4.5) imply the estimate of the controls in (4.4). To get the estimate for the states we observe that φ T =ŷ T −ȳ satisfies the equation where y T ,θ =ȳ + θ T (ŷ T −ȳ) with θ T : Q −→ [0, 1] measurable. Then, applying Theorem A.3 and Remark 5.2 we infer φ T W (0,∞) ≤ K 3 û T −ū L 2 (Q ω ) . Combining this estimate with the one established for the controls we deduced (4.4).
Funding Open access funding provided by University of Graz. The authors have not disclosed any funding.

Declarations
Competing interests The authors have not disclosed any competing interests.
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/.
Definition A. 1 We call y a solution to (A.1) if y ∈ L 2 loc (0, ∞; H 1 (Ω)), and for every T > 0 the restriction of y to Q T = Ω × (0, T ) belongs to W (0, T ) ∩ L ∞ (Q T ) and satisfies the following equation in the variational sense where M f is given by (1.5), K ∞ = y L ∞ (Q) , C K ∞ as in (1.6) with M = K ∞ , and Proof The existence and uniqueness of a solution y of (A.1) is a consequence of [10, Theorem 2.1]. Now, we assume that y ∈ L 2 (Q). The proof is split into several steps.
On the other hand, taking κ = 1 + 2α ( p−2) pn , r = 2 pκ α ( p−2) , and q = 2κ we get 1 r + n 2q = n 4 . Then, using [16,] we obtain the existence of a constant C 3 independent of T such that Then, we have Combining this inequality with (A.8) and (A.9) we deduce Let us estimate ξ 0 . For this purpose we distinguish two cases. First, we assume that n = 2 and p ∈ (2,4]. The last inequality follows from the fact y ∈ L 2 (Q) ∩ L ∞ (0, ∞; L 2 (Ω)) and 2 p α ( p−2) ≥ 2 because α ∈ (1, p 2 ) and p 2 ≤ p p−2 for p ≤ 4. For the remaining cases we observe that 2α ( p−2) p > 1 and additionally 2α ( p−2) p < 6 if n = 3. Now, we argue as follows we get with (A.10) Hence, |A 2ρ (t)| = 0 for almost every t ∈ (0, T ) holds. Since T > 0 was arbitrarily selected and all the constants above are independent of T , we deduce that |y(x, t)| ≤ 2ρ for almost all (x, t) ∈ Q and (A.5) follows with (A.4). Now, we explain the changes in the proof for the case n = 1. To get an analogous inequality to (A.8), we use the following Gagliardo-Nirenberg inequality see, for instance, [6, P. 233]. Then, we have with Hölder and Young inequalities From here we infer On the other side, we apply (A.9) with r = 8 and q = 4 and arguing as for the cases n = 2 or 3 we obtain To estimate ξ 0 we use again that y ∈ L 2 (Q) ∩ L ∞ (0, ∞; L 2 (Ω)) and proceed as follows The rest of the proof follows as for the cases n = 2 or 3.

Remark 5.1
The proof of the boundedness of y in Q follows some ideas of the proof of [16, Theorem III−7.1]. In that theorem, the boundedness is established for finite time horizon and the L ∞ (Q T ) estimates depend on time T . In our theorem, we have avoided the dependence with respect to time exploiting the fact that y ∈ L 2 (Q), which was used to estimate ξ 0 . By a simple modification of our proof, the L ∞ (Q) estimate of y can be also obtained in terms of g L r (0,∞;L q (Ω)) if 1 r + n 2q < 1. We observe that the assumption y ∈ L 2 (Q) is natural in the context of our optimal control problem due to the structure of its cost functional. Another difference of our estimates with respect to [16, Theorem III−7.1] concerns the choice of the boundary condition. Here we have treated the Neumann case while the Dirichlet case was considered in the mentioned reference. The only difference in our proof for the Dirichlet case consists in the definition of ρ that should include the L ∞ (Σ) norm of the Dirichlet datum, if it is not zero. Now, we analyze the following linear equation This implies Since z solves (A.12) in (0, T ε ), we have z ∈ W (0, T ε ) and z W (0,T ε ) can be estimated by h L 2 (0,∞;H 1 (Ω) * ) + z 0 L 2 (Ω) . This along with the above estimate implies the desired estimate of z in L 2 (0, ∞; H 1 (Ω)) ∩ L ∞ (0, ∞; L 2 (Ω)). From the equation (A.12) we infer that ∂z ∂t ∈ L 2 (0, ∞; H 1 (Ω) * ) and estimate (A.16) follows. Finally, under the additional regularity of h and z 0 , applying Theorem A.2 to the equation ∂z ∂t − Δz + az = g = h − d(x, t, y)z ∈ L p (0, ∞; L 2 (Ω)) ∩ L 2 (Q) with f = 0 and M f = 0 there, we infer that z ∈ L ∞ (Q) and (A.17) holds. Here we have used that L ∞ (0, ∞; L 2 (Ω)) ∩ L 2 (Q) ⊂ L p (0, ∞; L 2 (Ω)) for every p ≥ 2.
From (A.23) and the above estimate we deduce the boundedness of {ϕ T } T >0 in W (0, ∞). Then, there exists a sequence {T k } ∞ k=1 with T k → ∞ and a function ϕ ∈ W (0, ∞) such that ϕ T k ϕ in W (0, ∞) as k → ∞. It is obvious that we can pass to the limit in (A.23) and deduce that ϕ satisfies (A.20) and estimate (A.21) holds.