A nonlinear control law for a stochastic infinite time problem

The expectation of a particular class of nonquadratic performance criterion involving even powers of the state variables up to sixth order is minimized, over an infinite horizon, subject to a linear stochastic system. The process noise is composed of both additive white noise and state dependent white noise processes. The resulting controller is composed of a linear and cubic function of the state. Furthermore, this controller depends upon the noise variances of both the additive and state dependent noise processes. For partially observable systems with no state dependent noise, similar results as the completely observable system are implied by the separation theorem. For state dependent noise alone, a stochastic Lyapunov function is obtained from which simple probability bounds for the trajectory to exit from a given region of the state space are determined.