Optimal control problems for a von Kármán system with long memory

In this paper, we study quadratic cost optimal control problems governed by a von Kármán system with long memory. We prove the existence of an optimal control for the cost. Then, by proving the strong Gâteaux differentiability of nonlinear solution mapping we establish necessary optimality condition for the optimal control corresponding to the quadratic cost. Further, we study the time local uniqueness of the optimal controls for distributive observation.


Introduction
Let Ω be an open bounded domain in R 2 with a sufficiently smooth boundary ∂Ω. We set Q = ð0, TÞ × Ω, Σ = ð0, TÞ × ∂ Ω. We consider the following von Kárman system with long memory and the hinged boundary condition in the variables u and v, representing the deflection of the plate and the Airy's stress, respectively: where ′ = ∂/∂t, the vector ν denotes an outward normal, γ > 0 means a constant related to the rotational inertia, kð·Þ ∈ C 1 ð½0, TÞ is a memory kernel, f is a forcing function, and [·,·] is the von Kárman bracket given by The term qu in Equation (1) represents the reset force of the elastic plate in the system. This physical situation naturally leads to the consideration of the bilinear control problem for the control function q, which is used as a force to make the state close to a desired state taking into account. In this motivation, Bradley and Lenhart [1] studied the bilinear optimal control problem for a Kirchhoff plate equation (cf. [2]). And it has been studied in [3] the bilinear optimal control problem of velocity term in a Kirchhoff plate equation.
Motivated by [1,3] with the above physical background, we study here the bilinear minimax control problem for Equation (1) with the control function q based on the Fréchet differentiabilities of the nonlinear solution map. More detailed explanations are given as follows: In our previous study [4], we considered the Dirichlet boundary value problems of Equation (1) without the term qu and studied the optimal control problems for the external forcing control system by the frameworks in Lions [5]. In [4], we proved and used the Gâteaux differentiability of the nonlinear solution map to present the necessary optimality conditions for the optimal controls of the specific observation cases.
In this paper, we show the Fréchet differentiability of the solution map q ⟶ u from the bilinear control input terms to the solutions of Equation (1). In most cases, the Gâteaux differentiability may be enough to solve a quadratic cost optimal control problem. However, the Fréchet differentiability of a solution map is more desirable for studying the problem with more general cost function like nonquadratic or nonconvex functions. So, this study is an improvement on a previous study [4]. Based on the result, we constructed and solved the bilinear minimax optimal control problems in Equation (1). The minimax control strategies have been used by many researchers for various control problems (see Lasiecka and Triggiani [6] and Li and Yong [7]). As explained in [8], the minimax control framework is employed to take into account of the undesirable effects of system disturbance (or noise) in control inputs such that a cost function achieves its minimum even in the worst disturbances of the system. For the purpose, we replace the bilinear multiplier q in Equation (1) by c + η, where c is a control variable that belongs to the admissible control set C ad , and η is a disturbance (or noise) that belongs to the admissible disturbance set D ad . We also introduce the following cost function to be minimized within C ad and maximized within D ad : where uis a solution of Equation (1), z d ∈ L 2 ðQÞ is desired value, and the positive constants α and β are the relative weights of the second and third terms on the right hand side of (3). Our goal of this paper is to find and characterize the optimal control of the cost function (3) for the worst disturbance through control input in Equation (1).
This leads to the problem of finding and characterizing the saddle point (c * , η * ) ∈C ad × D ad satisfying In this paper, we use the terminology optimal pair for such a saddle point (c * , η * ) in (4). For the study of the existence of an optimal pair (c * , η * ) satisfying (4), we can find results in [8]. In that paper, the author used the minimax theorem in infinite dimensions given in Barbu and Precupanu [9]. And in [10], we extended the result to a quasilinear PDE.
On the other hand, in this paper, we use the method given in [11] to obtain the uniqueness as well as the existence of an optimal pair. That is to say, we use the strict convexity (or concavity) arguments of [12] by proving twice (Fréchet) differentiability of the solution map. Also, as we will see later, this method can suggest another condition that ensure strict convexity (or concavity) of the map from control (or noise) to the quadratic cost function (3).
Next, we derive an optimality condition for such a (c * , η * ) in (4). To derive the condition, we refer to the studies on bilinear optimal control problems where the state equations are linear partial differential equations such as the reaction diffusion equation or Kirchhoff plate equation (see [1,3,8,13] and references therein).
We now explain the content of this paper. In Section 2, we present notations and some necessary lemmas. In Section 3, we prove the well-posedness of Equation (1) with respect to u in the Hadamard sense using some previous results. To name just a few, we can refer to [14][15][16], and references therein. Especially, in order to prove the local Lipschitz continuity of the nonlinear solution map, we employ the energy equality of Volterra-type integro-differential equation which is proved in [17]. In Section 4, we shall study the minimax optimal control problems: at first, we shall show that the solution map of Equation (1): q ⟶ u is the first and twice Fréchet differentiable; By using twice Fréchet differentiability of the solution maps c ⟶ u and η ⟶ u, we prove that the maps c ⟶ J and η ⟶ J are strictly convex and concave, respectively, under the assumptions that α, β are sufficiently large or T > 0 is sufficiently small. And we also prove that the maps c ⟶ J and η ⟶ J are lower and upper semicontinuous, respectively. Consequently, we can prove the uniqueness and existence of an optimal pair. Next, we derive the necessary optimality condition of an optimal pair for the observation case associated with the cost (3).

Notations and Preliminaries
Throughout this paper, we use C as a generic constant and omit the integral variables in any definite integrals without confusion.
If X is a Banach space, we denote by X ′ its topological dual, and by h⋅ , ⋅i X ′,X the duality pairing between X ′ and X. We introduce the following abbreviations: where p ≥ 1 and W k,p is the L p -based Sobolev spaces for k ≥ 1. We denote by H k , the standard Sobolev spaces W k,2 for k ≥ 1. And H k 0 means the completions of C ∞ 0 ðΩÞ in H k for k ≥ 1. The duality pairs between H k 0 and H −k ðk = 1, 2Þ are abbreviated by h⋅ , ⋅ i k,−k . The scalar product and norm on L 2 are denoted by ð·, · Þ 2 and k⋅k 2 , respectively. Then, based on the Poincaré inequality and the well-known regularity theory for elliptic boundary value problems (Temam [18] p. 150), the scalar products on H k 0 (k = 1, 2) can be given as follows: Then obviously, We define the operator A which stands for the following: Journal of Function Spaces and consider the operator M γ = I + γA. We also define the operator A as follows: We note that By using again the well-known elliptic regularity theory (Temam [18] p. 150), we can obtain Therefore, we can employ It becomes apparent that each topological imbedding is continuous and compact. According to Adams [19], we know that when n ≤ 3, the imbedding is compact. It is well known that the biharmonic operator is bijective, and it admits an isometric extension Thus, we can define an operator G ∈ ℒ ðL 2 , Therefore, from Equation (1), one can also note that We collect below some results for the Airy stress function and von Kárman bracket. Consequently, Proof. See [15,20].

Well Posedness of a von Kárman Equation with Long Memory
We introduce the Hilbert space Wð0, TÞ of the weak solutions of Equation (1) given by with the norm 3 Journal of Function Spaces where D ′ ð0, TÞ is the space of distributions on ð0, TÞ.
As indicated in [14], von Kárman nonlinearity is subcritical; thus, the issues of well-posedness and regularity of weak solutions are standard.
, TÞ, q ∈ L ∞ ðQÞ, and f ∈ L 2 ðQÞ, then a weak solution u of Equation (1) exists and satisfies: To show the regularity of a weak solutions of Equation (1), we need the following lemma.

Corollary 6.
Assume that u is a weak solution of Equation (1). Then, we can assert (after possibly a modification on a set of measure zero) that Proof. From Dautray and Lions ([22], p. 480), it is clear that Wð0, TÞ↪Cð½0, T ; H 1 0 Þ ∩ C 1 ð½0, T ; L 2 Þ: Therefore, since u ∈ Wð0, TÞ ∩ L ∞ ð0, T ; VÞ ∩ W 1,∞ ð0, T ; H 1 0 Þ, the proof is the immediate consequence of Lemma 5 obtained by setting X = V, Y = H 1 0 to have u ∈ C w ð½0, T ; VÞ and by setting X = In the sequel, we give the important energy equality of weak solutions of Equation (1). It is used to prove the improved regularity of weak solutions of Equation (1) and used in all estimations in this paper.

Lemma 7.
Assume that u is a weak solution of Equation (1). Then, for each t ∈ ½0, T, we have the energy equality where Δv 0 = −Δ −1 ½u 0 , u 0 : Proof. By Corollary 6 and the uniform boundedness theorem, we have uðtÞ ∈ V and u ′ ðtÞ ∈ H 1 0 for all t ∈ ½0, T: Thus, every function in (30) has meaning for all t ∈ ½0, T: Then, we can proceed the proof as in ( [17], Proposition 2.1). By regarding f in ( [17], Proposition 2.1) as ½u, v + qu + f in Equation (1), we can deduce that the weak solution u of From [4], we can have Thus, we have (30). This proves the lemma.
From the energy equalities (30) or (31) together with the following well-known Gronwall's lemma, we can prove uniqueness and regularity of weak solutions of Equation (1).

Lemma 8.
Let ξð·Þ be a nonnegative, absolutely continuous function on ½0, T, which satisfies the differentiable inequality for a:e:t ∈ ½0, T: where ψ and ϕ are nonnegative, summable functions on ½0, T. Then for all 0 ≤ t ≤ T.
Here, we can state the following theorem.
Theorem 9. Assume that ðu 0 , u 1 Þ ∈ V × H 1 0 , k ∈ C 1 ð½0, TÞ, q ∈ L ∞ ðQÞ, and f ∈ L 2 ðQÞ. Then Equation (1)   Journal of Function Spaces Indeed, let p 1 = ðu 1 0 , u 1 1 , q 1 , f 1 Þ ∈ P and p 2 = ðu 2 0 , u 2 1 , q 2 , f 2 Þ ∈ P , we prove this theorem by showing the following inequality where C > 0 is a constant depending on the data and Proof of Theorem 9. Lemma 7 allows us to show the regularity of u. It is verified from the data conditions that the right hand side of (30) is continuous in t. Hence, we have that is continuous on ½0, T: Indeed, u ∈ Cð½0, T ; VÞ ∩ C 1 ð½0, T ; H 1 0 Þ: Therefore, considering results in [15,16] and [14], we can deduce that Equation (1) possesses a unique weak solution u ∈ Sð0, TÞ under the data condition ðu 0 , u 1 Based on the above result, we prove the inequality (35). For the purpose, we denote Then, we can know from Equation (1) that ϕ and Y satisfy the following equation in the weak sense: We note that Just as deriving the equality (31) from Equation (1), we can know that the weak solution ϕ of Equation (39) satisfies At first, we note by (13) that: By u 2 ∈ Sð0, TÞ↪L 2 ðQÞ and (38), we can get from (43) that For other estimates of the remaining terms on the right hand side of (41), we can refer to the previous results in [4] and obtain with (41) and (42)-(44) the following: And also, for almost t ∈ ½0, T, we obtain by Lemma 2 and (11) that By (38), (46), and (47), we can obtain This implies with (46) that Since M −1 γ A 1/2 ∈ LðL 2 Þ and M −1 γ ∈ LðL 2 , VÞ, by conducting similar estimations in Equation (39), we can obtain from (49) that Hence, by (49) and (50) we can prove (35). This completes the proof.

Quadratic Cost Minimax Control Problems
Let the following be the set of the admissible controls: where c and c are given constants, representing lower and upper bounds of the admissible control variables, respectively. Let the following be the set of the admissible disturbances or noises: where η and η are given constants, representing lower and upper bounds of admissible disturbance variables, respectively. For variational analysis, we use the L 2 ðQÞ norm on C ad and D ad : For simplicity, we denote by B ad = C ad × D ad : From Theorem 9, we can uniquely define the solution map B ad ⟶ Sð0, TÞ, which maps from q = ðc, ηÞ ∈ B ad to the weak solution uðqÞ ∈ Sð0, TÞ, where uðqÞ satisfies the following equation: The weak solution uðqÞ of Equation (53) is called the state of the control system Equation (53).
To study the quadratic cost minimax optimal control problems for Equation (53), we introduce the following quadratic cost function where z d ∈ L 2 ðQÞ is the desired value, and the positive constants α and β are the relative weights of the second and third terms on the right hand side of (54).
As indicated in the introduction, we shall study the minimax optimal control problem as follows: we prove the uniqueness as well as existence of a control c * ∈ C ad and a dis-turbance (or noise) η * ∈ D ad such that ðc * , η * Þ is a saddle point of the functional Jð·, · Þ of (54). That is, Here, we call such ðc * , η * Þ in (55) to be an optimal pair for the minimax optimal control problem with the cost (54). And we need to characterize ðc * , η * Þ in (55) by giving the necessary optimality condition through adjoint equation related to Equation (53) and the cost (54). For this purpose, we have to show the differentiabilities of the control to state mapping.

Differentiabilities of the Nonlinear Solution Map.
We study the Fréchet differentiability of the nonlinear solution map, which is an improvement of the previous results in Journal of Function Spaces [4] and is more desirable for many applications. From Theorem 9, for fixed ðu 0 , u 1 , f Þ ∈ V × H 1 0 × L 2 ðQÞ in Equation (53), we know that the solution map L ∞ ðQÞ ⟶ Sð0, TÞ from q ð = c + η in Equation (53)) ∈L ∞ (Q) to uðqÞ ∈ Sð0, TÞ is well defined and continuous.
For our study, we present the following definitions.
Definition 10. The solution map q ⟶ uðqÞ of L ∞ ðQÞ into Sð0, TÞ is said to be Fréchet differentiable on L ∞ ðQÞ if for any q ∈ L ∞ ðQÞ, there exists a TðqÞ ∈ LðL ∞ ðQÞ, Sð0, TÞÞ such that, for any w ∈ L ∞ ðQÞ, The operator TðqÞ is called the Fréchet derivative of u at q, which we denote by DuðqÞ:TðqÞw = DuðqÞw ∈ Sð0, TÞ is called the Fréchet derivative of u at q in the direction of w ∈ L ∞ ðQÞ: Definition 11. Let U be a subset of L ∞ ðQÞ and q, q * ∈ U. The solution map q ⟶ uðqÞ of U into Sð0, TÞ is said to be Gâteaux differentiable at q * in the direction q − q * if there exists a function Duðq * ; q − q * Þ ∈ Sð0, TÞ such that Theorem 12. The solution map q ⟶ uðqÞ of L ∞ ðQÞ into S ð0, TÞ is Fréchet differentiable on L ∞ ðQÞ and the Fréchet derivative of uðqÞ at q in the direction w ∈ L ∞ ðQÞ, that is to say z = DuðqÞw, is the weak solution of We prove this theorem by two steps.
(i) For any w ∈ L ∞ ðQÞ, Equation (58) admits a unique weak solution z ∈ Sð0, TÞ, namely, there exists an operator T ∈ LðL ∞ ðQÞ, Sð0, TÞÞ satisfying Tw = zð= zðwÞÞ (ii) We show that kuðq + wÞ − uðqÞ − zk Sð0,TÞ ≤ C kwk 2 L ∞ ðQÞ : Then, we can estimate the right hand side of (59) as follows. By (11) and Lemma 2 we have This implies with (38) that Similarly, we have Hence, by (61) and (62), we note that Taking into account wuðqÞ ∈ L 2 ðQÞ and (63), we can employ the linear theory in [17] (cf. [22]) to verify that Equation (58) admits a unique weak solution z ∈ Sð0, TÞ: And also by using the energy equality to be satisfied by z as in (31) and following similar estimations in Theorem 9, we can know by (38) that the weak solution zð= zðwÞÞ of Equation (58) satisfies Hence, from (64), the mapping w ∈ L ∞ ðQÞ ↦ zðwÞ ∈ Sð0, TÞ is linear and bounded. We can thus infer that there exists a T ∈ LðL ∞ ðQÞ, Sð0, TÞÞ such that Tw = zðwÞ for each w ∈ L ∞ ðQÞ: 7 Journal of Function Spaces (ii) We set the difference uðq + wÞ − uðqÞ − z = θ. Then, by noting the following: we know that θ satisfies in the weak sense, where In a similar argument to (63), we know that Thus with (68), we can apply the energy equality like (31) to (66) and follow similar estimations as in the Proof of Theorem 9, to obtain By Theorem 9 and (64), we can deduce as follows.
To show the uniqueness as well as existence of an optimal pair, we are going to use the strict convexity arguments in [12]. To this end, we consider the following results.
Proof. (i) By (38) and (64), we can have the following estimate Thus by similar arguments in the proof of (i) of Theorem 12, we can show that the weak solution ρ of Equation (75) can be estimated as follows: By (70) and (77), we know by (78) that (ii) From Equation (58), we can deduce that κ = Duðq + wÞw is the weak solution of the following equation: By previous result, we can verify the following From Equation (58), Equation (75) and Equation (80), χ = κ − z − ρ satisfies the following equation in the weak sense, where Þ , 9 Journal of Function Spaces where θ = uðq + wÞ − uðqÞ − z: By Theorem 9, (38) and (81), we can have By analogy with (84), we obtain Also, by analogy with (84), we have It holds by (77) that By (81) and Theorem 12, we can have By similar arguments to those in the Proof of Theorem 9, we can deduce that the weak solution χ of Equation (82) satisfies From (84) to (89), we can deduce which implies This completes the proof.

Corollary 14.
The map q ⟶ uðqÞ of U ⊂ L ∞ ðQÞ into Sð0, TÞ is twice Gâteaux differentiable at q * and such the twice Gâteaux derivative of u in the direction q − q * , say ρ = D 2 uðq * Þðq − q * , q − q * Þ, is a unique solution of Equation (75) in which q and w are replaced by q * and q − q * , respectively.
Proof. The proof is immediately followed by Theorem 13.
Proof. Let zðwÞ be the weak solution of Equation (58). Then, using Sð0, TÞ↪C 0 ð QÞ, we can get from (64) the following: Let ρ be the weak solution of Equation (75). From the second inequality in (77) and (78), we can deduce with (38) and (93) the following: This completes the proof.

Uniqueness and Existence of an Optimal Pair.
To study the existence of an optimal pair, we present the following results.
Proposition 16. The solution mapping q ⟶ uðqÞ from B ad to Wð0, TÞ of Equation (53) is continuous from the weaklystar topology of B ad to the weak topology of Wð0, TÞ: Before we prove Proposition 16, we need the following compactness lemma. Lemma 17. Let X, Y, and Z be Banach spaces such that the embeddings X ⊂ Y ⊂ Z are continuous and the embedding X ⊂ Y is compact. Then, a bounded set of W 1,∞ ð0, T ; X, ZÞ = fg | g ∈ L ∞ ð0, T ; XÞ, g′ ∈ L ∞ ð0, T ; ZÞg is relatively compact in Cð½0, T ; YÞ: