Optimal Controller for a Nonautonomous Linear Stochastic System with a Two-Sided Cost Functional

The stochastic linear control problem over an infinite-time horizon with a two-sided cost functional and a time-varying diffusion matrix is considered. In the two-sided quadratic cost functional, the limits of integration have opposite sign and depend on the length of planning horizon. It is shown that under conditions on the diffusion matrix growth, the well-known linear feedback law is optimal in terms of the extended long-run average cost and its pathwise analog. In addition, the probabilistic behavior of the system's optimal path is studied.