Existence of a Mean-Square Stabilizing Solution to a Modified Algebraic Riccati Equation

In this paper, we investigate a mean-square stabilizing solution to a modified algebraic Riccati equation (MARE), which arises in our previous work on the linear quadratic optimal control for linear time-invariant discrete systems with random input gains. An explicit necessary and sufficient condition ensuring the existence of a mean-square stabilizing solution is given directly in terms of the system parameters with the help of theory of positive operators and the assumption of observability or detectability of certain stochastic systems is no longer needed. Such a necessary and sufficient condition is compatible with that for the existence of a stabilizing solution to the standard definite algebraic Riccati equation (ARE).