The Optimal Control Problems for Generalized Elliptic Quasivariational Inequalities

Shih-Sen Chang,Min Liu,Salahuddin Salahuddin,Jinfang Tang,Abdullah Ali H Ahmadini

doi:10.3390/sym14020199

Shih-Sen Chang, Min Liu + Show 3 more

Open Access

https://doi.org/10.3390/sym14020199

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Abstract

In this article, we propose an optimal control problem for generalized elliptic quasi-variational inequality with unilateral constraints. Then, we discuss the sufficient assumptions that ensure the convergence of the solutions to the optimal control problem. The proofs depend on convergence results for generalized elliptic quasi-variational inequalities, obtained by the arguments of compactness, lower semi-continuity, monotonicity, penalty and different estimates. As an application, we addressed the abstract convergence results in the analysis of optimal control associated with boundary value problems.

Highlights

Many applications of optimal control theory can be found in physics, mechanics, automatics, systems theory, and financial management control theory
The study of optimal control problems for variational and hemivariational inequalities has been addressed in several works and is an expanding and vibrant branch of applied mathematics with numerous applications, see [11,12,13,14,15,16,17]
The theory and computational techniques for optimal control for equations and variational inequalities have been studied for quite some time

Summary

Introduction

Many applications of optimal control theory can be found in physics, mechanics, automatics, systems theory, and financial management control theory. We study, in this paper, the following optimal control problem for finding (x∗, f ∗) ∈ Vad such that. Consider a set K ⊂ X , an operator N : X × X −→ X and an element f ∈ Y With these data, we suggest the following perturbation of (1) for finding x ∈ Ksuch that. N(x, x), y − x X + j(x, y) − j(x, x) ≥ f, γy − γx Y , ∀y ∈ K. and, for a objective functional L : X × Y −→ R, we construct the following perturbation of the optimal control problem (3) for finding (x∗, f∗) ∈ Vad such that. We investigate and describe the applications in contact mechanics and a heat transfer process

Preliminaries

Main Results

Optimal Control Analysis

Optimal Control Associated with Frictional Contact Problem

A Stationary Heat Transfer Problem with Unilateral Constraints