Optimal control and stablilization for linear continuous-time mean-field systems with delay

TL;DR

This paper addresses optimal control and stabilization of continuous-time mean-field systems with input delay, establishing new solvability conditions and controller design methods that account for delays and multiplicative noise.

Contribution

It introduces the first necessary and sufficient conditions for optimal control with delay and derives controllers overcoming the separation principle obstacle.

Findings

Established solvability conditions for delayed mean-field control

Derived an optimal controller for systems with multiplicative noise

Proved stabilizability criteria via algebraic Riccati equation

Abstract

This paper studies optimal control and stabilization problems for continuous-time mean-field systems with input delay, which are the fundamental development of control and stabilization problems for mean-field systems. There are two main contributions: 1) To the best of our knowledge, the present paper is first to establish the necessary and sufficient solvability condition for this kind of optimal control problem with delay, and to derive an optimal controller through overcoming the obstacle that separation principle no longer holds for multiplicative-noise systems; 2) For the stabilization problem, under the assumption of exact observability, we strictly prove thatthe system is stabilizable if and only if the algebraic Riccati equation has a unique positive definite solution.