Controllability Gramian for Stochastic Game-Based Systems

This article is concerned with the exact controllability of a linear stochastic game-based control system (SGBCS) with time-variant random coefficients. Some necessary conditions and sufficient conditions for the exact terminal controllability of the SGBCS are obtained. Then, it is proved that the exact controllability of the SGBCS, the nonsingularity of a generalized version of the Gramian matrix, the validity of an observability inequality for the adjoint equation, and the unique solvability of a family of minimization problems are all equivalent. An admissible control for the regulator is constructed in closed form, which can steer the macrostate from any existing initial vector to any desired terminal random variable.