Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon

This paper is concerned with a stochastic linear-quadratic (LQ) problem in an infinite time horizon with multiplicative noises both in the state and the control. A distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite. A new type of algebraic Riccati equation - called a generalized algebraic Riccati equation (GARE) - is introduced which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints. It is then shown that the well-posedness of the indefinite LQ problem is equivalent to a linear matrix inequality (LMI) condition, whereas the attainability of the LQ problem is equivalent to the existence of a "stabilizing solution" to the GARE. Moreover, all possible optimal controls are identified via the solution to the GARE. Finally, it is proved that the solution to the GARE can be obtained via solving a convex optimization problem called semidefinite programming.