Indefinite Linear Quadratic Optimal Control for Discrete Time-Varying Stochastic Systems with Infinite Markov Jumps

In this note, we investigate the indefinite linear quadratic (LQ) optimization problem for discrete time-varying stochastic systems with infinite Markov jumps, where the weighting matrices of state and control in the cost function are allowed to be indefinite. The attainability of the indefinite LQ optimal control problem is equivalent to the solvability of coupled generalized algebraic Riccati equations, which is a countably infinite set, then the optimal solutions are given. Finally, as an important application, the Nash equilibrium problem is solved by the obtained indefinite LQ problem.