Optimization of fully observable nonlinear stochastic uncertain controlled diffusion: monotonicity properties and optimal sensitivity

This paper is concerned with fully observable nonlinear stochastically controlled diffusions, in which uncertainty is described by a relative entropy constraint between the nominal measure and the uncertain measure, while the pay-off is a functional of the uncertain measure. This is a minimax game, equivalent to the so-called nonlinear H^∞ optimal disturbance attenuation problem, 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. The objectives of this paper are twofold. First, to investigate the minimax problem in an abstract formulation, using its dual unconstrained functional. The dual formulation leads to several monotonicity properties of the optimal function, in terms of the nominal measure and an estimate of the uncertain measure. In addition the characterization is important for computing, as well as comparing, the solution of sub-optimal disturbance attenuation problems to the optimal one. Second, to apply the results of the abstract formulation to stochastic uncertain systems, in which the nominal and uncertain systems are described by conditional distributions. The results obtained include existence of the optimal control policy, explicit computation of the worst case conditional measure, and characterization of the optimal disturbance attenuation, for nonlinear systems.