Parametrizing stabilizing controls in stochastic systems

We consider discrete linear systems with noises that depend on both state and control. We give necessary and sufficient stabilizing conditions that yield a parametric description (parametrization) of all linear stabilizing controls with static output feedback that provide exponential stability in average of a quadratic closed system. We also give constructive sufficient conditions that allow computing the amplification matrix of a stabilizing control with an auxiliary optimization problem with constraints in the form of linear matrix inequalities without iterative procedures. The results extend to random structure systems and simultaneous stabilization problems. We show two possible applications of our results to robust control problems. The first application is a special case of the simultaneous stabilization problem, and the second is based on considering a stochastic model as a comparison system for a system with affine indeterminacies. We give an example of solving the simultaneous stabilization problem for a hypothetical aircraft.