Infinite horizon indefinite stochastic linear quadratic control for discrete-time systems

This paper discusses discrete-time stochastic linear quadratic(LQ)problem in the infinite horizon with state and control dependent noise,where the weighting matrices in the cost function are assumed to be indefinite.The problem gives rise to a generalized algebraic Riccati equation(GARE)that involves equality and inequality constraints.The well-posedness of the indefinite LQ problem is shown to be equivalent to the feasibility of a linear matrix inequality(LMI).Moreover,the existence of a stabilizing solution to the GARE is equivalent to the attainability of the LQ problem.All the optimal controls are obtained in terms of the solution to the GARE.Finally,we give an LMI-based approach to solve the GARE via a semidefinite programming.