Fréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problems
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.
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 , 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 , Spagnolo , Pohozaev , Lions , Nishihara and Yamada , 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 ). Based on this result, Hwang and Nakagiri  set up optimal control problems developed by Lions  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 ). 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 . 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 , 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 , Lasiecka and Triggiani , and Li and Yong .
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 , in which the author employed the minimax theorem in infinite dimensions given by Barbu and Precupanu . 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.
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 thatThe duality pairing between and is denoted by . It is clear thatEach space is dense in the following one, and the injections are continuous and compact. According to Adams , we know that the embeddingsare compact when .
The solution space of (1) of strong solutions is defined bywhich is endowed with the normwhere 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
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  and the previous result of , 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 , 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:whereIn estimating in (19), we can refer to the previous results [8, Theorem 2.1] to obtain the following inequality:Since and , we haveTogether with (21) and (22), we can deduce the following:Applying (23) to (19), we haveFrom (23) and (24), we can obtainThis 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 atThanks to the fact that and (10), we can know that . Thus we haveConsequently, 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 ofBased 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) LetThen from Theorem 4 and (14), we can estimate the above as follows:Hence, by (34) we know thatTo estimate the solution of (32), we take the scalar product of (32) with in Integrating (36) over , we obtainThe 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 obtainIn view of (32), (43) implies thatTherefore, from (43) and (44), we can know that , and the solution of (32) satisfiesHence, 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 satisfieswhere If we let then by similar arguments used for (34), we haveThanks to (50), if we follow similar arguments as in (i), then we can arrive atFrom (14), Theorem 4, and (45), we can deduce the following:Hence, from (51) to (54), we can obtainwhich 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 thatwhere 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 satisfieswhere 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 deduceWe can estimate of (57) as follows: