Optimization of stochastic uncertain systems with variational norm constraints

This paper considers optimization of stochastic uncertain systems on general abstract spaces, when the uncertainty of the system is described by a variational norm constraint. The pay-off is defined as a linear functional of the uncertain measure. By invoking the Hanh-Banach theorem, the maximization problem of the linear functional over the constraint set is shown to correspond to a convex combination of L₁ and L_infin norms. Further, the maximizing measure is constructed using a tilted exponential probability measure. The abstract results are subsequently employed to formulate a new class of uncertain continuous-time nonlinear stochastic controlled systems, in which the control seeks to minimize a linear functional while the measure seeks to maximize it over the variational norm constraint set. The variational norm uncertainty model admits uncertainties in both drift and diffusion coefficients of the stochastic differential equation describing the system. Hence it is much more general than existing uncertainty models described by relative entropy constraints.