On the Optimal Control Problem for a Linear Stochastic System with an Unstable State Matrix Unbounded at Infinity

Abstract

We consider a control problem over an infinite time horizon with a linear stochastic system with an unstable asymptotically unbounded state matrix. We extend the notion of anti-stability of a matrix to the case of non-exponential anti-stability, and introduce an antistability rate function as a characteristic of the rate of growth for the norm of the corresponding fundamental matrix. We show that the linear stable feedback control law is optimal with respect to the criterion of the adjusted extended long-run average. The designed criterion explicitly includes information about the rate of anti-stability and the parameters of the disturbances. We also analyze optimality conditions.