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

This paper discusses discrete-time stochastic linear quadratic (LQ) problem in theinfinite horizon with state and control dependent noise, where theweighting matrices in the cost function are assumed to beindefinite. The problem gives rise to a generalized algebraicRiccati equation (GARE) that involves equality and inequalityconstraints. The well-posedness of the indefinite LQ problem isshown to be equivalent to the feasibility of a linear matrixinequality (LMI). Moreover, the existence of a stabilizing solutionto the GARE is equivalent to the attainability of the LQ problem.All the optimal controls are obtained in terms of the solution tothe GARE. Finally, we give an LMI -based approach to solve the GAREvia a semidefinite programming.