OPTIMAL CONTROL OF A PARABOLIC EQUATION WITH MEMORY

. An optimal control problem for a semilinear parabolic partial diﬀerential equation with memory is considered. The well-posedness as well as the ﬁrst and the second order diﬀerentiability of the state equation is established by means of Schauder ﬁxed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The ﬁrst and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coeﬃcient of the operator. Finally, the suﬃciency of the second order optimality condition for the local optimal control is proved.


Introduction
In this paper, we study the following optimal control problem (P) min u∈L ∞ (0,T ;L 2 (Ω)) J(u) := 1 2 Q (y u − y d ) 2 dx dt + κ 2 Qω u 2 dx dt, where Q = Ω × (0, T ) with Ω a bounded domain of R n , 1 ≤ n ≤ 3, 0 < T < ∞, Q ω = ω × (0, T ), ω a measurable subset of Ω with positive measure, and y u is the solution of the following Neumann initial-boundary value problem ∂y ∂t − ∆y + f (x, t, y) + K[y] = g + χ ω u in Q, ∂ ν y = 0 on Σ, y(0) = y 0 in Ω. (1.1) In the above equation, K[y] denotes the function a(x, t, s, y(x, s)) dµ(s) for (x, t) ∈ Q, (1.2) where µ is a real Borel measure in [0, T ]. Such a term represent the memory. It is easy to understand that memory exists in almost all applications, in particular, in the diffusion processes described by parabolic partial differential equations. There are at least two commonly accepted situations that this will happen: (i) As it is known, the classical heat equation is derived from the Fourier's law. In the derivation, for simplicity, people neglect the memory/time delay effect. If one takes this into account, then the memory appears. See [15], as well as [20,27]. We see that the situation is actually more difficult since the memory can appear in the highest derivative terms. Here, we only consider a much simpler version. But it is still meaningful since we may regard it as an external heat source/sink depending on the past and up to current temperature. (ii) In diffusion of population/epidemic models, it is easy to understand that the current diffusion situation heavily depends on the past and up to current concentration of the spices. See [22], and references cited therein. By using the term K[y] is one way to describe such a situation. If we consider the Lebesgue decomposition of µ: dµ = hdt + σ, a typical form of the measure µ corresponds with the case where σ is a combination of Dirac measures, σ = In the above, the two terms on the right-hand side represent the continuous and the discrete memories, respectively. We point out that the diffusion process under consideration could have some special memory at some specific time moments. For example, suppose we are considering a heating process starting from certain initial temperature distribution y 0 (x). Then the changing of the temperature distribution at time t might be affected by the action made by the programmed machine at the previous moments t 1 , t 2 , · · · . One could easily cook up some other similar examples. The so-called fading memory is a main feature of the memory kernel a(· , · , · , ·). It plays an essential role for infinite horizon problems, and such a feature could be also interesting for finite horizon problems. The fading memory can be characterized by the following: Without the term K[y], the semi-linear parabolic equations have been extensively studied. See [19] for the standard classical theory, and [10,16,17] for some further/recent developments. The corresponding optimal control problems can be found, for instance, in [4,7,9] and the references therein. Parabolic equations with memory have been investigated by a number of authors for various situations [2,13,24,25]. There were some optimal control problems studied for the abstract evolution equations and some PDEs with memory, see [1,26]. However, it seems to us that the equation of form (1.1) has not been discussed and of course, the corresponding optimal control problem has not been touched. The purpose of the current paper is to analyze equations of the form (1.1), and carry out the corresponding optimal control theory.
The rest of the paper is organized as follows. In Section 2, we present a careful analysis on the state equation. It turns out that due to the appearance of the memory term K[y] governed by a general memory kernel and the general signed measure, together with the possibly super-linear growth of the nonlinear term f , the wellposedness of (1.1) becomes a little technically subtle. Optimal control problem is investigated in Section 3. It includes the existence of optimal controls, the first and the second order necessary conditions, and the sufficiency of the second order optimality condition for the local optimal control. We indicate that because of the memory term involves the real valued measure µ, the adjoint equation has a term of unknown function with µ as a part of the coefficient. This bring us a proper Bochner integral interpretation of the term, which makes the first and the second order necessary conditions very interesting and attractive. Some concluding remarks are collected in Section 4.

Analysis of the parabolic equation with memory
In this section, we perform the analysis of the following semilinear parabolic equation with memory: ∂ n y = 0 on Σ, y(0) = y 0 in Ω. (2.1) We make the following assumptions on the data of this equation.
(A4) The function a : Q × [0, T ] × R −→ R is measurable and of class C 2 with respect to the last variable and it satisfies: a(x, t, s, 0) = 0, (2.5) ∃ C a such that ∂a ∂y (x, t, s, y) + ∂ 2 a ∂y 2 (x, t, s, y) ≤ C a , for almost all (x, t) ∈ Q. Furthermore, we assume that a and ∂ j a ∂y j , j = 1, 2, are continuous with respect to the third variable. where |µ| denotes the total variation measure of µ; see, for instance, Chapter 6 of [21]. In the sequel, we will simply write µ . Given a function y : Ω × [0, T ] −→ R continuous with respect to the second variable, we set a(x, t, s, y(x, s)) dµ(s) for (x, t) ∈ Q.
Remark 2.1. We observe that the assumption (2.2) can be replaced by the more general hypothesis f (· , · , 0) ∈ L r (0, T ; L p (Ω)). Indeed, it is enough to rename f and g as f − f (· , · , 0) and g − f (· , · , 0), respectively. Analogously, we can relax the assumption (2.5). If we denote byâ : Q −→ R the function defined bŷ it is enough to assume thatâ ∈ L r (0, T ; L p (Ω)) and to replace a and g by a −â and g −â, and to define K[y] accordingly. The condition onâ holds if sup s∈[0,T ] a(· , · , s, 0) L r (0,T ;L p (Ω)) < ∞ is satisfied. Now, we address the issue of existence, uniqueness, and regularity of a solution to (2.2).
Theorem 2.2. Under the assumptions (A1)-(A5), (2.1) has a unique solution y ∈ W (0, T ) ∩ C(Q). In addition, there exist constants C W and C ∞ independent of (g, y 0 ) such that the following estimates are satisfied: Finally, if the weak convergence g k g in L r (0, T ; L p (Ω)) holds, then y k y in W (0, T ) and y k − y C(Q) → 0 as k → ∞, where y k and y are the states associated with g k and g, respectively.
Proof. Let {y 0,k } ∞ k=1 be a sequence of Lipschitz functions inΩ such that y 0,k C(Ω) ≤ y 0 C(Ω) and y 0,k − y 0 C(Ω) → 0 as k → ∞. Associated with k, we also define the functions a k (x, t, s, y) = a(x, t, s, Proj [−k,+k] (y)), where Proj [−k,+k] (y) = max{−k, min{y, +k}}, and a k (x, t, s, w(x, s)) dµ(s). From (2.5) and (2.6) we get |a k (x, t, s, y)| ≤ C a k and, consequently, for every Carathéodory function function w : Ω × [0, T ] −→ R. Now, we define the function F k : C(Q) −→ C(Q) by y k,w = F k (w) solution of the equation ∂ n y = 0 on Σ, y(0) = y 0,k in Ω. (2.10) Due to (2.9), we get that g − K k [w] ∈ L r (0, T ; L p (Ω)). Hence, (2.10) has a unique solution y k,w ∈ W (0, T ) ∩ C(Q) and it satisfies, for some α ∈ (0, 1] independent of w y k,w W (0,T ) ≤ C g L 2 (Q) + C k µ + y 0,k L 2 (Ω) , y k,w C 0,α (Q) ≤ C g L r (0,T ;L p (Ω)) + C k µ + y 0,k C 0,1 (Ω) ; see [4] or [7] for the existence and uniqueness of solutions and [11] for the Hölder estimate. Above, as along the proofs in this paper, C will denote a generic constant that could be different from line to line. Therefore, the image of F k is a bounded and closed subset of C 0,α (Q), hence it is a compact subset of C(Q). Then, applying Schauder's fixed point theorem we infer the existence of a function y k ∈ W (0, T ) ∩ C 0,α (Q) satisfying ∂ n y k = 0 on Σ, y k (0) = y 0,k in Ω. (2.11) Hence, we infer from the above inequalities and the fact that t was arbitrarily selected in (0, T ] Applying Gronwall's inequality to the function h(t) = max 0≤s≤t y k (s) 2 Therefore, the last estimates and the fact that y 0,k → y 0 in L 2 (Ω) yield Combining this estimate with (2.12), using again [4] or [7], and the fact that y 0,k L ∞ (Ω) ≤ y 0 L ∞ (Ω) , we deduce Hence, we have that K k [y k ] = K[y k ] for every k large enough. Using the above estimates and the fact that y 0,k → y 0 , it is easy to pass to the limit in (2.11) and to deduce that y k y in W (0, T ) and y is a solution of (2.1). Moreover, the estimates (2.7) and (2.8) are straightforward consequences of the estimates proved for {y k } ∞ k=1 . Let us prove the uniqueness of solution. Let y 1 , y 2 ∈ W (0, T ) ∩ C(Q) be solutions of (2.1) and set y = y 2 − y 1 . Then, subtracting the equations satisfied by y 2 and y 1 , we obtain with the mean value theorem ) with e −4Λ f t y, taking into account (2.6), and arguing similarly as above, we infer from the Gronwall inequality that y satisfies the inequality (2.13) with g = 0 and y 0 = 0 in the right hand side. Then, the equality y = 0 follows. Finally, we prove the continuity of the solution with respect to the right-hand-side of the equation. Let g k g in L r (0, T ; L p (Ω)) and denote by y k and y the solutions of (2.1) corresponding to g k and g, respectively. From (2.7) and (2.8) we know that {y k } ∞ k=1 is bounded in W (0, T ) ∩ C(Q). Therefore, for a subsequence, y k ỹ in W (0, T ). Since the embedding W (0, T ) ⊂ L 2 (Q) is compact and {y k } ∞ k=1 is bounded in C(Q), we can assume, taking a new subsequence if necessary, that y k (x, t) →ỹ(x, t) almost everywhere in Q and y k →ỹ strongly in L q (Q) for every q < ∞. Hence, it is easy to pass to the limit in the equation satisfied by y k and to deduce that y is the state associated to g. Therefore, the identityỹ = y follows and the whole sequence {y k } ∞ k=1 converges weakly to y in W (0, T ). Now, setting z k = y k − y we have is bounded in a Hölder space C 0,α (Q); see [11]. Therefore, the convergence z k → 0 strongly in C(Q) holds. Now, we define the mapping F : L r (0, T ; L p (Ω)) −→ W (0, T ) ∩ C(Q) by F (g) = y g solution of (2.1) associated with g. The next theorem analyzes the differentiability of F . First, we introduce the following notation. Given functions y, z, z 1 , 16) and z h1,h2 = F (g)(h 1 , h 2 ) satisfies equation Proof. We are going to apply the implicit function theorem. To this end, we define the space and endowed it with the norm Then, Y is a Banach space. We also define the function It is immediate to check that F is of class C 2 and F(F (g), g) = (0, 0) for every g ∈ L r (0, T ; L p (Ω)). Moreover, for y, z ∈ Y we have ∂F ∂y (y, g)z = ∂z ∂t − ∆z + ∂f ∂y (·, ·, y)z + K [y]z, z(0) .
Since has a unique solution z ∈ Y for every (h, z 0 ) ∈ L r (0, T ; L p (Ω)) × C(Ω). This property follows from Theorem 2.2. Indeed, if we definef (x, t, z) = ∂f ∂y (x, t, y g (x, t))z, b(x, t, s, z) = ∂a ∂y (x, t, s, y g (x, s))z, and b(x, t, s, z(x, s)) dµ(s), taking into account that y g ∈ C(Q), we infer thatf and b satisfy the assumptions (A3) and (A4), respectively. Then, Theorem 2.2 applies to the equation Therefore, from the implicit function theorem we deduce that F is of class C 2 and (2.16) and (2.17) follow by diferentiation of the identity F(F (g), g) = 0.
Before finishing this section we are going to carry out a more detailed study of the linearized equation (2.16).
Theorem 2.4. For every g ∈ L r (0, T ; L p (Ω)) and h ∈ L 2 (Q), the equation (2.16) has a unique solution z ∈ H 1 (Q) ∩ C([0, T ]; H 1 (Ω)). Further, there exists a constant C g depending of g, but independent of h, such that Let us mention that, given a function z ∈ C([0, T ]; L 2 (Ω)), the integral defining K [y g ]z is a Bochner integral and actually we have that K [y g ]z ∈ L ∞ (0, T ; L 2 (Ω)). Indeed, for every t ∈ [0, T ], we get with (2.6) Proof of Theorem 2.4. Let {h k } ∞ k=1 ⊂ L r (0, T ; L p (Ω)) be a sequence converging strongly to h in L 2 (Q). Denote by z k the solution of (2.16) corresponding to h k . Then, definingK[z] andf as we did at the end of the proof of Theorem 2.3 and applying Theorem 2.2 we infer the existence of a constant independent of k such that z k W (0,T ) ≤ C h k L 2 (Q) . Hence, {z k } ∞ k=1 is bounded in W (0, T ). By taking a subsequence, we obtain z k z in W (0, T ). Then, it is easy to pass to the limit in the equations satisfied by z k and to deduce that z is a solution of (2.16). Observe that W (0, T ) ⊂ C([0, T ]; L 2 (Ω)) and, hence, z ∈ C([0, T ]; L 2 (Ω)). Now, from the boundedness of y g and (2.20), the regularity z ∈ H 1 (Q) ∩ C([0, T ]; H 1 (Ω)) follows; see Section III-2 of [23]. Moreover, using that z W (0,T )) ≤ C h L 2 (Q) and the estimates of [23], the inequality (2.19) is obtained.

Optimal control problem
In this section, we analyze the control problem (P). We prove existence of a solution and derive first and second order optimality conditions. For this purpose we make the following assumptions: (A6) The target state y d belongs to L 2 (Q) and the coefficient κ in the cost functional is strictly positive.
Under the assumption (A7) we have that g + uχ ω ∈ L ∞ (0, T ; L 2 (Ω)). Then, we can use Theorem 2.2 with r = ∞ and p = 2 to deduce the existence and uniqueness of a solution y u ∈ W (0, T ) ∩ C(Q) for every control u ∈ L ∞ (0, T ; L 2 (ω)). Actually, the mapping G : L r (0, T ; L 2 (ω)) −→ W (0, T ) ∩ C(Q) associating to each control its corresponding state G(u) = y u is well defined if r > 4 4−n . Moreover, from Theorem 2.3 we get that G(u) = F (g + uχ ω ) is of class C 2 . We observe that z v = G (u)v is the solution of (2.16) with h = vχ ω . By the chain rule we infer that the cost functional J : L r (0, T ; L 2 (ω)) −→ R is also of class C 2 . The following theorem provides the expressions for the first and second derivatives of J.
Theorem 3.1. For every u, v, v 1 , v 2 ∈ L r (0, T ; L 2 (ω)) with r > 4 4−n the following identities hold Now, we have that for x ∈ Ω the mapping h(t) = T t ∂a ∂y (x, s, t, y u (x, t))ϕ(x, s) ds is continuous in [0, T ] due to the continuity of y u and the continuity of ∂a ∂y on the last two variables. Hence, for every function z ∈ C(Q) the following identities are fulfilled: ∂a ∂y (x, t, s, y u (x, s))z(x, s) dµ(s) ϕ(x, t) dt dx Regarding equation (3.3), we have to explain what we mean by a solution.

Lemma 3.4. Equation
(3.7) Proof. To prove the uniqueness is enough to show that ϕ = 0 is the unique solution of the homogeneous equation.
To establish this we select z as the solution of (2.18) with y g = y u and h(x, t) = sign(ϕ(x, t)). Then, z ∈ Z and (3.6) holds with right hand side equal to 0, which implies that ϕ = 0. To prove the existence of a solution we firstly consider the case where µ ∈ L ∞ (0, T ). For every integer k ≥ 1 we consider the equation ∂ n ϕ = 0 on Σ, ϕ(T ) = 0 in Ω Then following the lines of the proof of Theorem 2.2 it is easy to deduce the existence of a solution ϕ k ∈ W (0, T ) ∩ C(Q) to this equation. Further, the following inequality is satisfied: As in inequality (2.13), the constant C depends on µ L 1 (0,T ) . Using this estimate in (3.3) we infer that ∂f ∂y (x, t, y u )ϕ k + K [y u ] * ϕ k µ is uniformly bounded in L 2 (Q). Hence, we have that {ϕ k } ∞ k=1 is bounded in W (0, T ). Therefore, for a subsequence denoted by itself, we have that ϕ k ϕ in W (0, T ) and ϕ k → ϕ strongly in L 2 (Q). Whence, we can pass to the limits as k → ∞ in the equation satisfied by ϕ k and to get that ϕ solves (3.3). Moreover, the estimate (3.7) follows from (3.8).
Precisely, the unique delicate point to pass to the limit is in the integral To prove this we observe that the convergence µ k * µ in M [0, T ] implies the pointwise conver- . This combined with the weak convergence ϕ k ϕ in L 2 (Q) proves (3.9). Therefore, ϕ is solution of (3.3) and satisfies (3.7).
We continue this section by deriving the optimality conditions. Since (P) is not a convex problem, it is convenient to deal not only with global minimizers, but also with local minimizers. We will say thatū is a local minimizer or local solution of (P) in the L r (0, T ; L 2 (ω)) sense with r > 4 4−n if there exists ε > 0 such that J(ū) ≤ J(u) whenever u −ū L r (0,T ;L 2 (ω)) ≤ ε.
The next theorem establishes a sufficient condition for local optimality.
The presence of the Tikhonov regularizing term κ 2 u 2 L 2 (Qω) is crucial in the proof of the above theorem. When κ = 0, the second order analysis is more complicate; see [3,6].

Concluding remarks
We have presented a general theory of optimal control problem for a class of semilinear parabolic equations with a possibly super-linear nonlinearity and with a memory term governed by a general memory kernel and general real valued Borel measure. Here are some remarks that we would like to collect.
− The appearance of a memory term of the form (1.2) makes the well-posedness of the state equation, as well as that of the adjoint equations, technically difficult. A careful analysis involving the Bochner integral and some delicate regularity results for parabolic equations help us to overcome the difficult.
− It is possible to include control constraints in the control problem such as U ad = {u ∈ L 2 (0, T ; L 2 (ω)) : u(t) ∈ K ad }, where K ad is a closed, convex, and bounded subset of L 2 (ω). In this case, the existence of an optimal control and the first order optimality conditions can be easily obtained following the approach of this paper with obvious modifications. For the choices K ad = {v ∈ L 2 (ω) : v L 2 (ω) ≤ γ}, 0 < γ < ∞, K ad = {v ∈ L 2 (ω) : α ≤ v(x) ≤ β for a.a. x ∈ ω}, −∞ < α < β < ∞, the second order analysis can be performed by using the techniques of [5].