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

Jin-Soo Hwang

doi:10.1155/2020/1859736

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.

Highlights

Let Ω be an open bounded domain in R2 with a sufficiently smooth boundary ∂Ω
Motivated by [1, 3] with the above physical background, we study here the bilinear minimax control problem for Equation (1) with the control function q based on the Fréchet differentiabilities of the nonlinear solution map
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

Summary

Introduction

Let Ω be an open bounded domain in R2 with a sufficiently smooth boundary ∂Ω. We set Q = ð0, TÞ × Ω, Σ = ð0, TÞ × ∂ Ω. Motivated by [1, 3] with the above physical background, we study here the bilinear minimax control problem for Equation (1) with the control function q based on the Fréchet differentiabilities of the nonlinear solution map. We derive the necessary optimality condition of an optimal pair for the observation case associated with the cost (3)

Notations and Preliminaries

Well Posedness of a von Kárman Equation with Long Memory

Quadratic Cost Minimax Control Problems