Optimal Actuator Location of the Norm Optimal Controls for Degenerate Parabolic Equations

TL;DR

This paper investigates optimal actuator placement for degenerate parabolic equations to minimize control norms, transforming the problem into a game-theoretic framework and establishing the optimality of relaxed solutions.

Contribution

It introduces a novel formulation of the actuator placement problem as a zero-sum game and proves the relaxed solution's optimality for classical control problems.

Findings

Formulated the actuator location problem as a two-person zero-sum game.

Developed four equivalent formulations of the optimization problem.

Proved the relaxed problem's solution is optimal for the classical problem.

Abstract

This paper focuses on investigating the optimal actuator location for achieving minimum norm controls in the context of approximate controllability for degenerate parabolic equations. We propose a formulation of the optimization problem that encompasses both the actuator location and its associated minimum norm control. Specifically, we transform the problem into a two-person zero-sum game problem, resulting in the development of four equivalent formulations. Finally, we establish the crucial result that the solution to the relaxed optimization problem serves as an optimal actuator location for the classical problem.