International Journal of Differential Equations
Volume 2019 (2019), Article ID 3238462, 16 pages
https://doi.org/10.1155/2019/3238462

Research Article

Fréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problems

Jin-soo Hwang

Department of Mathematics Education, College of Education, Daegu University, Jillyang, Gyeongsan, Gyeongbuk, Republic of Korea

Correspondence should be addressed to Jin-soo Hwang; jshwang@daegu.ac.kr

Received 22 October 2018; Accepted 29 November 2018; Published 3 February 2019

Academic Editor: Salim Messaoudi

Copyright © 2019 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.

Abstract.

We consider a damped Kirchhoff-type equation with Dirichlet boundary conditions. The objective is to show the Fréchet differentiability of a nonlinear solution map from a bilinear control input to the solution of a Kirchhoff-type equation. We use this result to formulate the minimax optimal control problem. We show the existence of optimal pairs and find their necessary optimality conditions.

1. Introduction

Let be an open bounded set of with a smooth boundary . We set , for We consider a strongly damped Kirchhoff-type equation described by the following Dirichlet boundary value problem:

where , is the displacement of a string (or membrane), , is a forcing function, and is a bilinear forcing term, which is usually a bilinear control variable that acts as a multiplier of the displacement term. denotes the Euclidean norm on . As is well known by Kirchhoff [1], the nonlinear part of (1) represents an extension effect of a vibrating string (or membrane). Many kinds of Kirchhoff-type equations have been research subject of many researchers (see Arosio [2], Spagnolo [3], Pohozaev [4], Lions [5], Nishihara and Yamada [6], and references therein).

From a physical perspective, the damping of (1) represents an internal friction in an elastic string (or membrane) that makes the vibration smooth. Therefore, we can obtain the well-posedness in the Hadamard sense under sufficiently smooth initial conditions (see [7]). Based on this result, Hwang and Nakagiri [8] set up optimal control problems developed by Lions [9] with (1) using distributed forcing controls. They proved the Gâteaux differentiability of the quasilinear solution map from the control variable to the solution and applied the result to derive the necessary optimality conditions for optimal control in some observation cases.

It is important and challenging to extend the optimal control theory to practical nonlinear partial differential equations. There are several studies on semilinear partial differential equations (see [10]). Indeed, the extension of the theory to quasilinear equations is much more restrictive because the differentiability of a solution map is quite dependent on the model due to the strong nonlinearity. Only a few studies have investigated this topic (see [8, 11, 12]). Thus, the differentiability of a solution map in any sense is important to study optimal control or identification problems. In most cases, Gâteaux differentiability may be enough to solve a quadratic cost optimal control problem as in [8]. However, to study the problem in more general cost function like nonquadratic or nonconvex functions, the Fréchet differentiability of a solution map is more desirable.

In this paper, we show the Fréchet differentiability of the solution map of (1): from the bilinear control input variables to the solutions of (1). In the author’s knowledge, the Fréchet differentiability of a quasilinear solution map is not studied yet. Based on the result, we construct and solve a bilinear minimax optimal control problem on (1). For the study, we refer to the linear results from Belmiloudi [13], in which the author considered some linear parabolic partial differential equations as the state equations for the problem. Minimax control framework has been used by many researchers for various control problems. There are many literatures related to the minimax control problems. We can refer to just a few: Arada and Raymond [14], Lasiecka and Triggiani [15], and Li and Yong [16].

In this paper, the minimax control framework was employed to take into account the undesirable effects of system disturbance (or noise) in control inputs such that a cost function achieves its minimum even when the worst disturbances of the system occur. For this purpose, we replace the bilinear multiplier in (1) by , where is a control variable that belongs to the admissible control set and is a disturbance (or noise) that belongs to the admissible disturbance set . We introduce the following cost function to be minimized within and maximized within :

where is a solution of (1), is a Hilbert space of observation variables, is an operator from the solution space of (1) to , is a desired value, and the positive constants and are the relative weights of the second and third terms on the RHS of (2).

As mentioned, another goal of this paper is to find and characterize the optimal controls of the cost function (2) for the worst disturbances through control input in (1). This leads to the problem of finding the saddle points of the cost function (2). First, we prove the existence of an admissible control and disturbance (or noise) such that is a saddle point of the functional of (2). That is,

Secondly, we derive an optimality condition for in (3). In this paper, we use the terminology optimal pair to represent such a saddle point in (3). To prove the existence of an optimal pair satisfying (3), we follow the arguments given by Belmiloudi [13], in which the author employed the minimax theorem in infinite dimensions given by Barbu and Precupanu [17]. Next, we derive the necessary optimal conditions for some observation cases that should be satisfied by the optimal pairs in these observation cases. To derive these conditions, we refer to the studies about bilinear optimal control problems where the state equation is linear partial differential equations such as the reaction diffusion equation or Kirchhoff plate equation (see [13, 18–20] and references therein).

We now explain the content of this paper. In Section 2, we prove the well-posedness of (1) in the Hadamard sense under sufficiently smooth initial conditions, including a stability estimate from the data space to the solution space. In Section 3, we shall show that the solution map of (1): is Fréchet differentiable. In Section 4, we shall study the minimax optimal control problems: By using the Fréchet differentiability of the solution maps and , we prove that the maps and are convex and concave, respectively, under the assumptions that are sufficiently large. And with an assumption on the operator in (2), we prove the maps and are lower and upper semicontinuous, respectively. As a result, we can prove the existence of an optimal pair. Next, we derive the necessary optimal conditions for some practical observation cases by employing associate adjoint systems. Especially, we use a first-order Volterra integrodifferential equation as a proper adjoint equation in the velocity’s observation case, which is another novelty of this paper.

2. Preliminaries

Throughout this paper, we use as a generic constant. Let be a Banach space. We denote its topological dual as and the duality pairing between and by . We also introduce the following abbreviations:

where . is the completions of in for . Let the scalar product on be . From Poincare’s inequality and the regularity theory for elliptic boundary value problems (cf. Temam [21, p. 150]), the scalar products on and can be endowed as follows:

Then we know that

The duality pairing between and is denoted by . It is clear that

Each space is dense in the following one, and the injections are continuous and compact. According to Adams [22], we know that the embeddings

are compact when .

The solution space of (1) of strong solutions is defined by

which is endowed with the norm

where and denote the first and second order distributional derivatives of

Definition 1. A function is said to be a strong solution of (1) if and satisfies

From Dautray and Lions [23, p.480] and Lions and Magnes [24], we remark that

The following variational formulation is used to define the weak solution of (1).

Definition 2. A function is said to be a weak solution of (1) if and satisfies

The following is the well-known Gronwall inequality.

Lemma 3. Let be a nonnegative, absolutely continuous function on , which satisfies the following differentiable inequality for a.e.

where and are nonnegative, summable functions on . Then

Proof. See Evans [25, p.624].

Throughout this paper, we will omit writing the integral variables in the definite integral without any confusion. Referring to [7] and the previous result of [8], we can obtain the following theorem on existence, uniqueness, and regularity of a solution of (1).

Theorem 4. Assume that , and Then (1) has a unique strong solution . Moreover, the solution mapping of into is locally Lipschitz continuous. Let and The following is satisfied:

where is a constant depending on the data.

Proof. From [7], for each fixed in (1), we can infer that (1) admits a unique strong solution under the data condition
Based on this result, for each and , we prove the inequality (18). For that purpose, we denote by . Then, from (1), we can know that satisfies the following:

where

In estimating in (19), we can refer to the previous results [8, Theorem 2.1] to obtain the following inequality:

Since and , we have

Together with (21) and (22), we can deduce the following:

Applying (23) to (19), we have

From (23) and (24), we can obtain

This completes the proof.

Corollary 5. For , the following inequality is satisfied:

where is a constant depending on the data and and are the solutions of (1) corresponding to and , respectively.

Proof. We denote by . Then, as in the proof of Theorem 4, we can know that satisfies the following:

where is given in (20). Estimating in (27) as in the proof of Theorem 4, we can arrive at

Thanks to the fact that and (10), we can know that . Thus we have

Consequently, from (28) and (29), we have (26).
This completes the proof.

3. Fréchet Differentiability of the Nonlinear Solution Map

In this section, we study the Fréchet differentiability of the nonlinear solution map. The Fréchet differentiability of the solution map plays an important role in many applications. Let We consider the nonlinear solution map from to , where is the solution of

Based on Theorem 4, for fixed , we know that the solution map , which maps from the term of (30) to , is well defined and continuous. We define the Fréchet differentiability of the nonlinear solution map as follows.

Definition 6. 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 , and is called the Fréchet derivative of at in the direction of

Theorem 7. The solution map of to is Fréchet differentiable on and the Fréchet derivative of at in the direction , that is to say , is the solution of

We prove this theorem by two steps:

(i)For any , (32) admits a unique solution That is, there exists an operator satisfying .(ii)We show that as

Proof. (i) Let

Then from Theorem 4 and (14), we can estimate the above as follows:

Hence, by (34) we know that

To estimate the solution of (32), we take the scalar product of (32) with in

Integrating (36) over , we obtain

The right hand side of (37) can be estimated as follows:

Considering (38)-(41) and taking , we can obtain the following from (37):

Applying Lemma 3 to (42), we obtain

In view of (32), (43) implies that

Therefore, from (43) and (44), we can know that , and the solution of (32) satisfies

Hence, from (45), the mapping is linear and bounded. From this, we can infer that there exists such that for each
(ii) We set the difference . Then, from (30) and (32), we can have the following:

Thus, we know from (46) that satisfies

where

If we let

then by similar arguments used for (34), we have

Thanks to (50), if we follow similar arguments as in (i), then we can arrive at

From (14), Theorem 4, and (45), we can deduce the following:

Hence, from (51) to (54), we can obtain

which immediately implies that as
This completes the proof.

The following result plays an important role in proving the existence of optimal controls in the next section.

Proposition 8. Given , the Fréchet derivative is locally Lipschitz continuous on with topology. Indeed, it is satisfied that

where is a constant depending on the data.

Proof. Let , be the solutions of (32) corresponding to , and we set . Then, by similar calculations as in (46), we can deduce that satisfies

where

By similar arguments as in the proof of (i) of Theorem 7, in (57) can be estimated as follows:

From Theorem 4, the embedding , and the first inequality of (45), we can deduce

We can estimate of (57) as follows:

From (61) to (64), we can obtain the following from (59):

This completes the proof.

4. Quadratic Cost Minimax Control Problems

In this section, we study the quadratic cost minimax optimal control problems for a damped Kirchhoff-type equation. Let the following be the set of the admissible controls:

Let the following be the set of the admissible disturbance or noises:

To perform our variational analysis, norms of and are preferable, even though and are subsets of For simplicity, let be a product space defined by

Using Theorem 4, we can uniquely define the solution mapping , which maps the term to the solution , which satisfies the following equation:

The solution of (68) is the state of the control system (68). From Theorem 7, we can deduce that the map of to is Fréchet differentiable at , and the Fréchet derivative of at in the direction , say is a unique solution of the following problem:

The quadratic cost function associated with the control system (68) is

where is a Hilbert space of observation variables, the operator is an observer, is a desired value, and the positive constants and are the relative weights of the second and the third terms on the RHS of (70).

To pursue our objective, we assume that the observer in (70) is a compact operator. As mentioned in the introduction, the minimax optimal control problem can be summarized as follows:

(i)Find an admissible control and a noise (or disturbance) such that is a saddle point of the functional of (70). That is,

(ii)Characterize (optimality condition).

Such a pair in (71) is called an optimal pair (or an optimal strategy pair) for the problem (70).

4.1. Existence of Optimal Pairs

To study the existence of optimal pairs, we present the following results.

Proposition 9. The solution mapping from to is continuous from the weakly-star topology of to the weak topology of

In proving the Proposition 9, we need the following compactness lemma.

Lemma 10. Let and be Banach spaces such that the embeddings are continuous and the imbedding is compact. Then a bounded set of is relatively compact in

Proof. See Simon [26].

Proof of Proposition 9. Let and let be a sequence such that

For simplicity, we let each state be a solution of

We conduct the scalar product of (73) with in

which immediately implies

The integration of (75) over implies

where

By conducting similar calculations to the proof of (i) of Theorem 7, we can obtain the following from (76):

Since we know from Theorem 4 that , we can note that

From (78) and (79), we can infer

Applying Lemma 3 to (80), we have

Theorem 4 and (81) imply that remains in a bounded set of Therefore, by using Rellich’s extraction theorem, we can find a subsequence of also called , and find such that

Since the embedding is compact, we can apply Lemma 10 to (83) and (84) with and in Lemma 10 to verify that

Hence, we can find a subsequence if necessary such that

Therefore, (82) and (86) imply

From (72) and (85), we can also extract a subsequence, if necessary, denoted again by such that

We replace by , if necessary, and take in (73). Then, by the standard argument in Dautray and Lions [23, pp.561-565], we conclude that the limit is a solution of

Moreover, from the uniqueness of solutions of (89), we conclude that in , which implies that weakly in .
This completes the proof.

We now study the existence of optimal pairs.

Theorem 11. Let the observer in (70) be a compact operator. Then, for sufficiently large and in (70), there exists such that satisfies (71).

Proof. Let be the map and let be the map To obtain the existence of optimal pairs in the minimax control problem, we follow the steps given by [13]: We prove that is convex and lower semicontinuous for all and that is concave and upper semicontinuous for all Then, we employ the minimax theorem in infinite dimensions (see Barbu and Precupanu [17]).
For sufficiently large and in (70), we first prove the convexity of and the concavity of . To prove the convexity of , which is a differentiable map, it is sufficient to show that

From Fréchet differentiability of the solution map , where is fixed, (90) can be rewritten as

where are solutions of (69), in which is replaced by , respectively. We can easily deduce that (91) is equivalent again to

From Corollary 5, Proposition 8, and (60), we can estimate the left hand side of (92) as follows:

Considering from (92) to (94), we can deduce that there exists a sufficiently large such that, for any , (92) holds true. Therefore, the map is convex.
Similarly, we can also show that there exist a sufficiently large such that the following inequality is satisfied for any :

This also indicates the concavity of
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 defined by (66) is a closed, bounded, and convex in , we can extract a subsequence such that

Then, by Proposition 9, we have ,

Thus, by the assumption that is a compact operator, we can extract a subsequence of , if necessary, denoted again by , such that

From (97), it can be easily verified for the same subsequence in (97) that

Due to the weakly lower semicontinuity in the norm topology, we can determine from (99) and (100) that the map is lower semicontinuous for all By similar arguments, we can prove that is upper semicontinuous for all
Hence, we know that

But since , we have

Similarly, we also know that there exists such that

From (102) and (103), we can conclude that is an optimal pair for the cost (70).
This completes the proof.

4.2. Necessary Conditions of Optimal Pairs

We now turn to the necessary optimality conditions that have to be satisfied by optimal pairs with the cost (70). For this purpose, we consider the following two types of observations , of distributive and terminal values:

(1)we take and and observe ;(2)we take and and observe

Remark 12. Clearly, the embedding is compact. From the embedding (14) we can utilize Lemma 10 in which and to obtain the embedding is also compact. Consequently, the observer is a compact operator. Thus, satisfies the requirement for the existence of optimal pairs given in Theorem 11.

Remark 13. Since , and the embedding is compact, we can employ the Aubin-Lions-Temam’s compact embedding theorem (cf. Temam [27, p. 274]) to determine that the embedding is compact. Consequently, the observer is a compact operator. Therefore, satisfies the requirement for the existence of optimal pairs given in Theorem 11.

4.2.1. Case of Distributive and Terminal Values Observations

In this observation case, we consider the cost function associated with the control system (68):

where and are desired values, and the positive constants and are the relative weight of the second and the third terms on the RHS of (104).

Now we formulate the following adjoint equation to describe the necessary optimality conditions for this observation:

where is defined in (33). Using a similar estimation to (34), we can have

Remark 14. By considering the observation conditions and and (106), we can refer to the well-posedness result of Dautray and Lions [23, pp.558-570] to verify that (105), reversing the direction of time , admits a unique weak solution , which is given in Definition 2.

We now discuss the first-order optimality conditions for the minimax optimal control problem (71) for the quadratic cost function (104).

Theorem 15. If and in the cost (104) are large enough, then an optimal control and a disturbance , namely, an optimal pair satisfying (71), can be given by

where is the weak solution of (105).

Proof. Let be an optimal pair in (71) with the cost (104) and let be the corresponding weak solution of (68).
From Theorem 7, we know that the map is Fréchet differentiable at in the direction , which satisfies for sufficiently small Thus, the map is also (strongly) Gâteaux differentiable at in the direction . Thus, we have

where is a unique solution of (69). Therefore we can obtain the Gâteaux derivative of the cost (104) at in the direction as follows:

where is a solution of (69).
Before we proceed to the calculations, we note that

We multiply both sides of the weak form of (105) by , which is a solution of (69), and integrate it over . Then, we have

By integration by parts and the terminal value of the weak solution of (105), (111) can be rewritten as

Since is the solution of (69), we can obtain the following from (112):

Therefore, we can deduce that (109) and (113) imply

Since is an optimal pair in (71), we know that

Therefore, we can obtain the following from (114) and (115):

where By considering the signs of the variations and in (116), which depend on and , respectively, we can deduce the following from (116) (possibly not unique):

This completes the proof.

4.2.2. Case of Velocity Observation

In this observation case, we consider the cost function associated with the control system (68):

where is a desired value and the positive constants and are the relative weight of the second and the third terms on the RHS of (118). Now we turn to the necessary optimality conditions that have to be satisfied by each solution of the minimax optimal control problem with the cost (118). For this purpose, as proposed in a previous study [8], we introduce the following adjoint equation corresponding to (68), in which is replaced by :

where is defined in (33).

Remark 16. Usually, adjoint systems of second order problems are also second order (cf. Lions [9]) as long as they are meaningful. However, we have a barrier in this quasilinear (68). If we derive a formal second order adjoint system related to the velocity observation with the cost (118), then it is hard to explain the well-posedness. To overcome this difficulty, we follow the idea given in [8, 11], in which it is adopted that the first-order integrodifferential system as an appropriate adjoint system instead of the formal second order adjoint system.

Proposition 17. Equation (119) admits a unique weak solution satisfying

where is the solution space of (119) given by

Proof. Since

the time reversed equation of (119) ( in (119)) is given by

where . From (106) and , it is verified that all requirements of Dautray and Lions [23, pp.656-661] are satisfied with (123). Therefore, it readily follows that there exists a unique weak solution of (123).
This completes the proof.

We now discuss the first-order optimality conditions for the minimax optimal control problem (71).

Theorem 18. If and in the cost (118) are large enough, then an optimal control and a disturbance , namely, an optimal pair satisfying (71), can be given by:

where is the weak solution of (119).

Proof. Let be an optimal pair in (71) with the cost (118) and be the corresponding weak solution of (68).
By analogy with the proof of Theorem 15, the Gâteaux derivative of the cost (118) at in the direction that satisfies for sufficiently small is given by

where is a solution of (69). We multiply both sides of the weak form of (119) by and integrate it over . Then, we have

By integration by parts and the terminal value of the weak solution of (119), (126) can be rewritten as

Since is the solution of (69), we can obtain the following from (127):

Therefore, we can deduce that (125) and (128) imply

Since is an optimal pair in (71), we know that

Therefore, we can obtain the following from (129) and (130):

where By considering the signs of the variations and in (131), which depend on and , respectively, we can deduce from (131) that (possibly not unique)

This completes the proof.

5. Conclusion

The Fréchet differentiability from a bilinear control input into the solution space of a damped Kirchhoff-type equation is verified. As an application of this result, we proposed a minimax optimal control problem for the above state equation by using quadratic cost functions that depend on control and disturbance (or noise) variables. By utilizing the Fréchet differentiability of the solution map and the continuity of the solution map in a weak topology, we have proven existence of the optimal control of the worst disturbance, called the optimal pair under some hypothesis. And we derived necessary optimality conditions that any optimal pairs must satisfy in some observation cases.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

The author read and approved the final manuscript.

Acknowledgments

This research was supported by the Daegu University Research Grant 2015.

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