Lie Algebraic Methods in Optimal Control of Stochastic Systems with Exponential-of-integral Cost

The purpose of this paper is to formulate and study the optimal control of partially observed stochastic systems with exponential-of-integral-sample cost, known as risk-sensitive problems, using Lie algebraic tools. This leads to the introduction of the su cient statistic algebra, Ls, through which one can determine a priori the maximum order of the controller. When dim(Ls) < 1, the construction of the control laws is addressed through extensions of the Wei-Norman method, as in nonlinear ltering problems. Aside from speci c known nite-dimensional examples which are studied in order to delineate the application of the Lie algebraic tools, new classes of nite-dimensional controllers are identi ed as well. In addition, relations with minimax dynamic games are explored to best assess the importance and generality of the nite-dimensional control systems.