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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

From a physical perspective, the damping of (

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 [

In this paper, we show the Fréchet differentiability of the solution map of (

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

As mentioned, another goal of this paper is to find and characterize the optimal controls of the cost function (

We now explain the content of this paper. In Section

Throughout this paper, we use

The solution space

A function

From Dautray and Lions [

The following variational formulation is used to define the

A function

The following is the well-known Gronwall inequality.

Let

See Evans [

Throughout this paper, we will omit writing the integral variables in the definite integral without any confusion. Referring to [

Assume that

From [

Based on this result, for each

This completes the proof.

For

We denote

This completes the proof.

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

The solution map

The operator

The solution map

We prove this theorem by two steps:

For any

We show that

(i) Let

(ii) We set the difference

This completes the proof.

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

Given

Let

This completes the proof.

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:

Using Theorem

The quadratic cost function associated with the control system (

To pursue our objective, we assume that the observer

Find an admissible control

Characterize

Such a pair

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

The solution mapping from

In proving the Proposition

Let

See Simon [

Let

This completes the proof.

We now study the existence of optimal pairs.

Let the observer

Let

For sufficiently large

Similarly, we can also show that there exist a sufficiently large

Next, we prove the existence of an optimal pair

Hence, we know that

This completes the proof.

We now turn to the necessary optimality conditions that have to be satisfied by optimal pairs with the cost (

we take

we take

Clearly, the embedding

Since

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

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

By considering the observation conditions

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

If

Let

From Theorem

Before we proceed to the calculations, we note that

This completes the proof.

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

Usually, adjoint systems of second order problems are also second order (cf. Lions [

Equation (

Since

This completes the proof.

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

If

Let

By analogy with the proof of Theorem

This completes the proof.

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.

No data were used to support this study.

The author declares no conflicts of interest.

The author read and approved the final manuscript.

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

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