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.


Introduction
Let Ω be an open bounded set of R  ( ≤ 3) with a smooth boundary Γ.We set  = (0, ) × Ω, Σ = (0, ) × Γ for  > 0. 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),  > 0,  is a forcing function, and U 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 R  .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 2 International Journal of Differential Equations 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): U →  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 U in (1) by  + V, where  is a control variable that belongs to the admissible control set U  and V is a disturbance (or noise) that belongs to the admissible disturbance set V  .We introduce the following cost function to be minimized within U  and maximized within V  : where  is a solution of (1),  is a Hilbert space of observation variables, C 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  * ∈ U  and disturbance (or noise) V * ∈ V  such that ( * , V * ) is a saddle point of the functional (, V) of (2).That is, Secondly, we derive an optimality condition for ( * , V * ) in (3).In this paper, we use the terminology optimal pair to represent such a saddle point ( * , V * ) in (3).To prove the existence of an optimal pair ( * , V * ) 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][19][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): U →  is Fréchet differentiable.In Section 4, we shall study the minimax optimal control problems: By using the Fréchet differentiability of the solution maps  →  and V → , we prove that the maps  →  and V →  are convex and concave, respectively, under the assumptions that ,  are sufficiently large.And with an assumption on the operator C in (2), we prove the maps  →  and V →  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 firstorder Volterra integrodifferential equation as a proper adjoint equation in the velocity's observation case, which is another novelty of this paper.

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  ≥ 1.   0 is the completions of  ∞ 0 (Ω) in   for  ≥ 1.Let the scalar product on  2 be (⋅, ⋅) 2 .From Poincare's inequality and the regularity theory for elliptic boundary value problems (cf.Temam [21, p. 150]), the scalar products on  1 0 and (Δ) =  2 ∩  1 0 can be endowed as follows: Then we know that The duality pairing between  1 0 and  −1 is denoted by ⟨, ⟩ 1,−1 .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  ≤ 3.
The solution space (0, ) of (1) of strong solutions is defined by which is endowed with the norm         (0,) where   and   denote the first and second order distributional derivatives of .
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).
Throughout this paper, we will omit writing the integral variables in the definite integral without any confusion.

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 F =  ∞ ().We consider the nonlinear solution map from  ∈ F to () ∈ (0, ), where () is the solution of Based on Theorem 4, for fixed ( 0 ,  1 , ) ∈ (Δ)× 1 0 × 2 (), we know that the solution map F → (0, ), which maps from the term  ∈ F of (30) to () ∈ (0, ), is well defined and continuous.We define the Fréchet differentiability of the nonlinear solution map as follows.Definition 6.The solution map  → () of F into (0, ) is said to be Fréchet differentiable on F if for any  ∈ F there exists a () ∈ L(F, (0, )) such that, for any  ∈ F, The operator () is called the Fréchet derivative of  at , which we denote by (), and () = () ∈ (0, ) is called the Fréchet derivative of  at  in the direction of  ∈ F. Theorem 7. The solution map  → () of F to (0, ) is Fréchet differentiable on F and the Fréchet derivative of () at  in the direction  ∈ F, that is to say  = (), is the solution of We prove this theorem by two steps: (i) For any  ∈ F, (32) admits a unique solution  ∈ (0, ).That is, there exists an operator  ∈ L(F, (0, )) satisfying  = (= ()).
Then from Theorem 4 and ( 14), we can estimate the above as follows: ≤ (with ( 14) and (8)) Hence, by (34) we know that To estimate the solution  of (32), we take the scalar product of (32) with −Δ  − Δ in  2 : Integrating (36) over [0, ], we obtain The right hand side of (37) can be estimated as follows: 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 ‖‖ (0,) = (‖‖ F ) as ‖‖ F → 0. 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  ∈ F, the Fréchet derivative () is locally Lipschitz continuous on F with  2 () topology.Indeed, it is satisfied that where  > 0 is a constant depending on the data.
Proof.Let   = (  ), ( = 1, 2) be the solutions of (32) corresponding to   , ( = 1, 2), and we set  =  1 −  2 .Then, by similar calculations as in (46), we can deduce that  satisfies where By similar arguments as in the proof of (i) of Theorem This completes the proof.

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,  2 () norms of U  and V  are preferable, even though U  and V  are subsets of F. For simplicity, let F  be a product space defined by Using Theorem 4, we can uniquely define the solution mapping F  → (0, ), which maps the term  = (, V) ∈ F  to the solution () ∈ (0, ), which satisfies the following equation:   () − (1 +     ∇ () The solution () of ( 68) is the state of the control system (68).
From Theorem 7, we can deduce that the map  = (, V) → () of F  to (0, ) is Fréchet differentiable at  =  * = ( * , V * ), and the Fréchet derivative of () at  =  * in the direction  = (ℎ, ) ∈ F 2 , 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 C ∈ L((0, ), ) 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 C(∈ L((0, ), )) 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  * ∈ U  and a noise (or disturbance) V * ∈ V  such that ( * , V * ) is a saddle point of the functional (, V) of (70).That is, (ii) Characterize ( * , V * ) (optimality condition).
Such a pair ( * , V * ) in ( 71) is called an optimal pair (or an optimal strategy pair) for the problem (70).

Existence of Optimal Pairs.
To study the existence of optimal pairs, we present the following results.Proposition 9.The solution mapping from F  to (0, ) is continuous from the weakly-star topology of F  to the weak topology of (0, ).
Proof of Proposition 9. Let  = (, V) ∈ F  and let   = (  , V  ) ∈ F  be a sequence such that   ⇀  weakly-star in F  as  → ∞. (72) For simplicity, we let each state   = (  ) be a solution of We conduct the scalar product of (73) with −Δ From ( 72) and (85), we can also extract a subsequence, if necessary, denoted again by   ≡ (  , V  ) 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 (0, ), which implies that (  ) ⇀ () weakly in (0, ).This completes the proof.
We now study the existence of optimal pairs.Theorem 11.Let the observer C in (70) be a compact operator.Then, for sufficiently large  and  in (70), there exists Proof.Let P V be the map  → (, V) and let Q  be the map V → (, V).To obtain the existence of optimal pairs in the minimax control problem, we follow the steps given by [13]: We prove that P V is convex and lower semicontinuous for all V ∈ V  and that Q  is concave and upper semicontinuous for all  ∈ U  .Then, we employ the minimax theorem in infinite dimensions (see Barbu and Precupanu [17]).
Similarly, we can also show that there exist a sufficiently large   (P, F  ,   , C) such that the following inequality is satisfied for any  >   (P, F  ,   , C): This also indicates the concavity of Q  .
Next, we prove the existence of an optimal pair ( * , V * ) ∈ F  by verifying that P V is lower semicontinuous for all V ∈ V  and Q  is upper semicontinuous for all  ∈ U  .Let {  } ⊂ U  be a minimizing sequence of .Thus Since U  defined by (66) is a closed, bounded, and convex in F, we can extract a subsequence {   } ⊂ {  } such that Then, by Proposition 9, we have weakly in  (0, ) as  → ∞.
(98) Thus, by the assumption that C ∈ L((0, ), ) is a compact operator, we can extract a subsequence of {   }, if necessary, denoted again by {   }, such that ∀V ∈ V  .From (97), it can be easily verified for the same subsequence {   } in (97) that Due to the weakly lower semicontinuity in the  2 () norm topology, we can determine from (99) and (100) that the map P V :  → (,V) is lower semicontinuous for all V ∈ V  .By similar arguments, we can prove that Q  is upper semicontinuous for all  ∈ U  .Hence, we know that But since  0 (V) ≤ ( * , V), we have Similarly, we also know that there exists V * ∈ V  such that From ( 102) and ( 103), we can conclude that ( * , V * ) ∈ F  is an optimal pair for the cost (70).This completes the proof.
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  * ∈ U  and a disturbance V * ∈ V  , namely, an optimal pair  * = ( * , V * ) ∈ F  satisfying (71), can be given by where  is the weak solution of (105).

Case of Velocity
Observation  2 .In this observation case, we consider the cost function associated with the control system (68): where   ∈  2 () 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  = (, V) is replaced by  * = ( * , V * ): where G(⋅, ⋅) 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.
We now discuss the first-order optimality conditions for the minimax optimal control problem (71).
Proof.Let  * = ( * , V * ) ∈ F  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  * = ( * , V * ) in the direction  = (ℎ, ) ∈ F 2 that satisfies  * +  ∈ F  for sufficiently small  > 0 is given by where  = ( * ) is a solution of (69).We multiply both sides of the weak form of (119) by   and integrate it over This completes the proof.

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.