Optimal Actuator Location of the Minimum Norm Controls for Stochastic Heat Equations

TL;DR

This paper investigates optimal actuator placement for controlling stochastic heat equations, formulating a game-theoretic approach to identify locations that minimize control effort while ensuring approximate controllability.

Contribution

It introduces a novel game-theoretic framework for optimal actuator placement in stochastic heat equations, linking relaxed solutions to classical control problems.

Findings

Established a Nash equilibrium condition for optimal actuator location.

Proved the relaxed solution is optimal for the classical problem.

Developed a sufficient and necessary condition for optimality.

Abstract

In this paper, we study the approximate controllability for the stochastic heat equation over measurable sets, and the optimal actuator location of the minimum norm controls. We formulate a relaxed optimization problem for both actuator location and its corresponding minimum norm control into a two-person zero sum game problem and develop a sufficient and necessary condition for the optimal solution via Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.