Small Noise Ergodic Dynamical Systems

Abstract

This chapter focuses on the small noise ergodic dynamical systems. It presents some recent results in small noise problems in the ergodic case and some possible implications for small noise ergodic control problems. It also presents an assumption in which Y0(x) is the optimal feedback control in the infinite-time deterministic control problem. Use of Y0 in the corresponding ergodic control problem (minimization of the cost function with respect to the resulting ergodic measure) yields a cost of dɛ + o(ɛ) for some constant d. Under certain assumptions, one could expect that the optimal cost would also satisfy dɛ + o(ɛ). Y0 would be a reasonably good suboptimal control in the ergodic problem for sufficiently small ɛ. The ergodic problems are also studied under assumptions that included periodicity of the state equations and cost function in the state variables.