Existence of a mean-square stabilizing solution to a modified algebraic Riccati equation

In this paper, the existence of a mean-square stabilizing solution to a discrete-time modified algebraic Riccati equation (MARE), which arises in the study of some stochastic linear-quadratic optimal control problem, is investigated. The theory of cone-invariant operators is employed as the mathematical tool to tackle this problem. We provide some criteria for the cone-stability, cone-observability and cone-detectability of a class of cone-invariant systems in terms of the associated distinguished eigenvalues. Then two scenarios are considered concerning the problem of existence. We first study the MARE under the assumption that the input weighting matrix in the cost function is positive definite. Then an explicit necessary and sufficient condition is obtained. It is compatible with the one ensuring the existence of a stabilizing solution to the standard algebraic Riccati equation. Such a condition is derived for the very first time and it indicates that the common condition of observability or detectability of certain stochastic system is unnecessary. Only the observability of the distinguished eigenvalue at 1 of an associated cone-invariant operator is required. We also generalize this result to a class of MAREs. However, when the input weighting matrix is only positive semi-definite, this condition does not hold. In this case, we get a sufficient condition and a necessary condition, respectively. These two conditions coincide when the input weighting matrix is indeed positive definite.