Exact Controllability for Mean-Field Type Linear Game-Based Control Systems

TL;DR

This paper investigates the exact controllability of mean-field game-based control systems generated by linear-quadratic Nash games, providing criteria and explicit controls for steering system states.

Contribution

It introduces Gram-type and Kalman-type criteria for controllability of mean-field game systems and establishes their equivalence with observability, including explicit control construction.

Findings

Derived Gram-type criterion for time-varying coefficients

Established Kalman-type criterion for time-invariant systems

Constructed explicit admissible controls for state steering

Abstract

Motivated by the self-pursuit of controlled objects, we consider the exact controllability of a linear mean-field type game-based control system (MF-GBCS, for short) generated by a linear-quadratic (LQ, for short) Nash game. A Gram-type criterion for the general timevarying coefficients case and a Kalman-type criterion for the special time-invariant coefficients case are obtained. At the same time, the equivalence between the exact controllability of this MF-GBCS and the exact observability of a dual system is established. Moreover, an admissible control that can steer the state from any initial vector to any terminal random variable is constructed in closed form.