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

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 time-varying 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.