Characterization of the optimal disturbance attenuation for nonlinear stochastic uncertain systems

This paper is concerned with an abstract formulation of stochastic optimal control systems, in which uncertainty is described by a relative entropy constraint between the nominal measure and the uncertain measure, while the payoff is a functional of the uncertain measure. This is a minimax game in which the controller seeks to minimize the pay-off, while the disturbance described by a set of measures aims at maximizing the pay-off. When stochastic dynamical systems are considered, this problem is known to be equivalent to the optimal disturbance attenuation problem arising in H^∞ control. The objective of this paper is to derive several properties of the optimal solution. One such property is the characterization of the optimal disturbance attenuation, in terms of the nominal measure and an estimate of the uncertain measure. The characterization is important for computing, as well as comparing, the solution of sub-optimal disturbance attenuation problems to the optimal one.