Bilinear Minimax Optimal Control Problems for a von Kárman System with Long Memory

Hwang, Jin-soo

doi:https://doi.org/10.1155/2020/1859736

Journal of Function Spaces

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Research Article | Open Access

Volume 2020 | Article ID 1859736 | https://doi.org/10.1155/2020/1859736

Bilinear Minimax Optimal Control Problems for a von Kárman System with Long Memory

Jin-soo Hwang¹

Academic Editor: Gisele Mophou

Received18 Feb 2020

Accepted12 May 2020

Published16 Jun 2020

Abstract

In the present article, we consider a von Kárman equation with long memory. The goal is to study a quadratic cost minimax optimal control problems for the control system governed by the equation. First, we show that the solution map is continuous under a weak assumption on the data. Then, we formulate the minimax optimal control problem. We show the first and twice Fréchet differentiabilities of the nonlinear solution map from a bilinear input term to the weak solution of the equation. With the Fréchet differentiabilities of the control to solution mapping, we prove the uniqueness and existence of an optimal pair and find its necessary optimality condition.

1. Introduction

Let be an open bounded domain in ² with a sufficiently smooth boundary . We set . We consider the following von Kárman system with long memory and the hinged boundary condition in the variables and , representing the deflection of the plate and the Airy’s stress, respectively: where , the vector denotes an outward normal, means a constant related to the rotational inertia, is a memory kernel, is a forcing function, and [·,·] is the von Kárman bracket given by

The term 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 , 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 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 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 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 in Equation (1) by , where is a control variable that belongs to the admissible control set , and η is a disturbance (or noise) that belongs to the admissible disturbance set . We also introduce the following cost function to be minimized within and maximized within : where is a solution of Equation (1), is desired value, and the positive constants and 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 () satisfying

In this paper, we use the terminology optimal pair for such a saddle point () in (4). For the study of the existence of an optimal pair () 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 () 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 [14–16], 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): is the first and twice Fréchet differentiable; By using twice Fréchet differentiability of the solution maps and , we prove that the maps and are strictly convex and concave, respectively, under the assumptions that , are sufficiently large or is sufficiently small. And we also prove that the maps and 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).

2. 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 is a Banach space, we denote by its topological dual, and by the duality pairing between and . We introduce the following abbreviations: where and is the -based Sobolev spaces for . We denote by , the standard Sobolev spaces for . And means the completions of in for . The duality pairs between and are abbreviated by . The scalar product and norm on are denoted by and , 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 () can be given as follows:

Then obviously,

We define the operator which stands for the following: and consider the operator . We also define the operator 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 , 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 by

Therefore, from Equation (1), one can also note that

We collect below some results for the Airy stress function and von Kárman bracket.

Lemma 1. The trilinear form : given by satisfies the property

Proof. See ([20], Proposition 1.4.2).

Lemma 2. The bilinear forms from into and from into are continuous. We also have the following estimates: Consequently,

Proof. See [15, 20].

3. Well Posedness of a von Kárman Equation with Long Memory

We introduce the Hilbert space of the weak solutions of Equation (1) given by with the norm

Definition 3. Function is called a weak solution of Equation (1), if it satisfies where is the space of distributions on .
As indicated in [14], von Kárman nonlinearity is subcritical; thus, the issues of well-posedness and regularity of weak solutions are standard.

Theorem 4. If , and , then a weak solution of Equation (1) exists and satisfies: To show the regularity of a weak solutions of Equation (1), we need the following lemma.

Lemma 5. Let be two Banach spaces, with dense, and being reflexive. Set Then

Proof. See ([21], p. 275).

Corollary 6. Assume that 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 Therefore, since , the proof is the immediate consequence of Lemma 5 obtained by setting to have and by setting to have .

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 is a weak solution of Equation (1). Then, for each , we have the energy equality where

Proof. By Corollary 6 and the uniform boundedness theorem, we have and for all Thus, every function in (30) has meaning for all Then, we can proceed the proof as in ([17], Proposition 2.1). By regarding in ([17], Proposition 2.1) as in Equation (1), we can deduce that the weak solution of Equation (1) satisfies 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 be a nonnegative, absolutely continuous function on which satisfies the differentiable inequality for where and are nonnegative, summable functions on . Then for all .

Proof. See ([23], p.624).

Here, we can state the following theorem.

Theorem 9. Assume that , and . Then Equation (1) has a unique weak solution in . Moreover, the solution mapping of into is locally Lipschitz continuous.
Indeed, let and , we prove this theorem by showing the following inequality where is a constant depending on the data and

Proof of Theorem 9. Lemma 7 allows us to show the regularity of . It is verified from the data conditions that the right hand side of (30) is continuous in . Hence, we have that is continuous on Indeed,

Therefore, considering results in [15, 16] and [14], we can deduce that Equation (1) possesses a unique weak solution under the data condition such that

Based on the above result, we prove the inequality (35). For the purpose, we denote by and by . Then, we can know from Equation (1) that and 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 of Equation (39) satisfies

At first, we note by (13) that:

By 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:

By applying Lemma 8 to (45), we have

And also, for almost we obtain by Lemma 2 and (11) that

By (38), (46), and (47), we can obtain

This implies with (46) that

Since and 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.

4. Quadratic Cost Minimax Control Problems

Let the following be the set of the admissible controls: where and 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 and are given constants, representing lower and upper bounds of admissible disturbance variables, respectively. For variational analysis, we use the norm on and For simplicity, we denote by From Theorem 9, we can uniquely define the solution which maps from to the weak solution where satisfies the following equation:

The weak solution 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 is the desired value, and the positive constants and 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 and a disturbance (or noise) such that is a saddle point of the functional of (54). That is,

Here, we call such in (55) to be an optimal pair for the minimax optimal control problem with the cost (54). And we need to characterize 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.

4.1. 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 [4] and is more desirable for many applications. From Theorem 9, for fixed in Equation (53), we know that the solution map from in Equation (53)) (Q) to is well defined and continuous.

For our study, we present the following definitions.

Definition 10. The solution map of into is said to be Fréchet differentiable on if for any , there exists a such that, for any The operator is called the Fréchet derivative of at , which we denote by is called the Fréchet derivative of at in the direction of

Definition 11. Let be a subset of and . The solution map of into is said to be Gâteaux differentiable at in the direction if there exists a function such that

Theorem 12. The solution map of into is Fréchet differentiable on and the Fréchet derivative of at in the direction , that is to say , is the weak solution of We prove this theorem by two steps. (i)For any Equation (58) admits a unique weak solution namely, there exists an operator satisfying (ii)We show that

Proof. (i) Let 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 and (63), we can employ the linear theory in [17] (cf. [22]) to verify that Equation (58) admits a unique weak solution And also by using the energy equality to be satisfied by as in (31) and following similar estimations in Theorem 9, we can know by (38) that the weak solution of Equation (58) satisfies Hence, from (64), the mapping is linear and bounded. We can thus infer that there exists a such that for each
(ii) We set the difference . Then, by noting the following: we know that 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. By Lemma 2 and (11), we have From (38), (64) and Theorem 9, we can obtain with (71) that By analogy with (71) and (72), we can deduce Hence, from (69) to (73), we can obtain

This completes the proof.

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.

Theorem 13. The map of into is twice Fréchet differentiable at and the twice Fréchet derivative of at in the direction , say , is a unique solution of the following problem where is the solution of Equation (58), is given in (59), and To prove Theorem 13, it is sufficient to show the following: (i);(ii).

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 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 is the weak solution of the following equation: By previous result, we can verify the following From Equation (58), Equation (75) and Equation (80), satisfies the following equation in the weak sense, where where 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 of Equation (82) satisfies From (84) to (89), we can deduce which implies

This completes the proof.

Corollary 14. The map of into is twice Gâteaux differentiable at and such the twice Gâteaux derivative of in the direction , say , is a unique solution of Equation (75) in which and are replaced by and , respectively.

Proof. The proof is immediately followed by Theorem 13.

Lemma 15. The weak solution of Equation (75) satisfies

Proof. Let be the weak solution of Equation (58). Then, using we can get from (64) the following: Let 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.

4.2. 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 from to of Equation (53) is continuous from the weakly-star topology of to the weak topology of

Before we prove Proposition 16, we need the following compactness lemma.

Lemma 17. Let , and be Banach spaces such that the embeddings are continuous and the embedding is compact. Then, a bounded set of is relatively compact in

Proof. See Simon [24].

Proof of Proposition 16. Let and let be a sequence such that From now on, each state is the weak solution of If we let and , then from Theorem 9, we can know that in Equation (96) and in and deduce that the following is fulfilled: Hence, we can deduce from (97) that and remain in the bounded sets of and , respectively. Therefore, by using Rellich's extraction theorem, we can find a subsequence of , say again , and find such that Since is compact, we can apply Lemma 17 to (99) and (100) with and to verify that Hence, we may extract a subsequence, denoted again by , such that By Lemma 2, (11) and (38), we can note for almost that Therefore, by (102) and (103), we may extract a subsequence, denoted again by , such that Since we can deduce with (104) that as For any , we consider for almost all that By Lemma 1, we note that the right hand side of (106) equals By considering we have . And through (14) we can know that and therefore Thus, by Lemma 2, (109), (11), and (38), we can have as follows: Thus, from (102), (105), and (110), we may extract a subsequence, if necessary, denoted again by , such that From (95) and (101), we can obtain We replace by in Equation (96), if necessary, and take . Then, by the standard arguments in Dautray and Lions ([22], pp.561-565), we conclude that the limit is a weak solution of Moreover, from the uniqueness of the weak solutions, we conclude that in , which implies that weakly in .

This completes the proof.

We now study the uniqueness and existence of an optimal pair.

Theorem 18. For sufficiently large and in (54) there exists unique such that satisfies (55).
We prove this Theorem by showing: (i)For sufficiently large and in (54), we prove the maps and are strictly convex and strictly concave, respectively(ii)We prove the existence of an optimal pair by showing the maps and are lower semicontinuous and upper semicontinuous, respectively

Proof. (i) Let be the map and let be the map To obtain the unique existence of an optimal pair in the minimax optimal control problem, we show the map is strictly convex and lower semicontinuous for all and the map is strictly concave and upper semicontinuous for all As in [11] (cf. [12]), to show the strict convexity and the strict concavity of each map, we verify the following second Gâteaux derivative conditions where For sufficiently large in (54), we first prove the convexity of by showing (114). For simplicity, we denote and by , and , respectively. We calculate From Corollary 14 we know that the map is twice Gâteaux differentiable at in the direction Thus from (116), we can obtain the second Gâteaux derivative of as follows: where is the weak solution of Equation (75) in which and are replaced by and , respectively. Then, by Lemma 15 and (117), we can deduce that Hence, we can verify that there exists sufficiently large such that (114) is satisfied for any Therefore, the map is strictly convex for sufficiently large .
Similarly, we can show that there exist a sufficiently large such that (115) is satisfied for any . This also indicates the strict concavity of
(ii) Next, we prove the existence of an optimal pair by verifying that is lower semicontinuous for all and is upper semicontinuous for all Let be a minimizing sequence of . Thus Since is bounded in , we can extract a subsequence such that Then, by Proposition 16, we obtain Since we know from Dautray-Lions ([22], p.480) that and the embedding is compact, we can use Lemma 17 in which to have from (121) that the sequence is relatively compact in . Thus, we can choose a subsequence of , if necessary, still denoted by such that Therefore, we can deduce for the same subsequence given above that Since the norm is weakly lower semicontinuous, we can verify by (120) and (123) that the map is lower semicontinuous for all Hence, we know that But since we have By similar arguments, we can prove that is upper semicontinuous for all Thus, we can also know that there exists such that From (125) and (126), we can conclude that is an optimal pair for the cost (54).

This completes the proof.

Remark 19. Assuming that is sufficiently small, instead of assuming and are large enough, we can also obtain strict convexity and strict concavity of the maps and , respectively.

4.3. Necessary Condition of an Optimal Pair

In this subsection, we study the necessary optimality condition to be satisfied by an optimal pair of the minimax optimal control problem with the cost (54).

From Theorem 12, we know that the solution map from to is Fréchet differentiable at and the Fréchet derivative of at in the direction say is a unique weak solution of the following equation: where is defined in (59). By Equation (127), we introduce the following adjoint equation corresponding to the cost function (54):

Proposition 20. Equation (128) admits a unique solution

Proof. Changing the variable to in Equation (128), we can complete the proof by referring to the results in [4, 17].
We now discuss the first-order optimality conditions for the minimax optimal control problem (55) for the quadratic cost function (54).

Theorem 21. If and in the cost (54) are large enough or is sufficiently small, then an optimal control and a disturbance , namely, an optimal pair satisfying (55) can be given by: where is the weak solution of Equation (128), and are constants given in (51), and and are constants given in (52).

Proof. Let be an optimal pair in (55) with the cost (54) and be the corresponding weak solution of Equation (53).
From Theorem 12, it is clear that the map is Gâteaux differentiable at in the direction with for sufficiently small Indeed, we have where is a unique weak solution of Equation (127). Therefore, by (130), we can get the Gâteaux derivative of the cost (54) at in the direction as follows: where is a solution of Equation (127).
Before we proceed to the calculations, by making use of the fact that is a self adjoint operator, we can note by Lemma 1 and (109) that We multiply both sides of the weak form of Equation (128) by which is a solution of Equation (127), and integrate it over . Then, we have By Fubini's theorem, we can know that From (132), (134) and terminal values of the weak solution of Equation (128), we can obtain the following by integration by parts of (133): Since is the solution of Equation (127), we can obtain the following from (135): Therefore, we can deduce that (131) and (136) imply Since is an optimal pair in (55), we know Therefore, we can obtain the following from (137) and (138): where By considering the signs of the variations and in (139), which depend on and , respectively, we can deduce from (139) that

This completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Daegu University Research Grant 2017.

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Copyright © 2020 Jin-soo Hwang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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