A Class of Multi-Objective Control Problems for Quasi-linear Parabolic Equations

TL;DR

This paper investigates a multi-objective hierarchical control problem for multi-dimensional quasi-linear parabolic equations, addressing controllability with complex coefficients involving the state and its gradient.

Contribution

It introduces a novel controllability framework for quasi-linear parabolic equations with coefficients depending on both the state and its gradient, extending previous results to higher spatial dimensions.

Findings

Established controllability results for systems with gradient-dependent coefficients.

Extended existing theories to higher spatial dimensions.

Provided a new approach to Nash equilibrium in hierarchical control.

Abstract

This paper is devoted to studying a multi-objective control problem for a class of multi-dimensional quasi-linear parabolic equations. The considered system is driven by a leader control and two follower controls. For each leader control, a pair of follower controls is searched for as a Nash quasi-equilibrium (or Nash equilibrium) of cost functionals, while the aim for a leader control is to solve a controllability problem. This hierarchic control problem may be transformed into controllability of a strongly coupled system of quasi-linear parabolic equations through one control. Regarding controllability for quasi-linear parabolic equations of second order, the existing results usually require coefficients in principal parts to be independent of gradient of solutions, or spacial dimension to be limited. In this paper, the coefficients in principal parts for the controlled quasi-linear system contain not only the state itself but also its gradient with general spacial dimension.