Minimum principle for partially observable nonlinear risk-sensitive control problems using measure-valued decompositions

Abstract

This paper is concerned with the derivation of a minimum principle for partially observed controlled diffusions with correlation between signals and observation noises, when the sample cost criterion is an exponential of integral. Instead of considering the information state which satisfies a version of Zakai's equation, measure-valued decompositions are employed to convert this equation into a P-a.s. deterministic equation, thus avoiding the basic finite dimensional approximation approach. Using weak control variations the minimum principle is derived. It consists of a modified version of Zakai's equation, an adjoint process satisfying a stochastic partial differential equation with terminal condition, and a Hamiltonian functional. This minimum principle is then applied to solve the linear-exponential-quadratic-Gaussian trackmgjproblem. The results of this paper are important in relating H ∞ or robust control problems and risk-sensitive control problems