Lie algebraic methods in optimal control of stochastic systems with exponential-of-integral cost

Abstract

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 sufficient statistic algebra, L s , through which one can determine á priori the maximum order of the controller. When dim( L s)<∞ , the construction of the control laws is addressed through extensions of the Wei–Norman method, as in nonlinear filtering problems. Aside from specific known finite-dimensional examples which are studied in order to delineate the application of the Lie algebraic tools, new classes of finite-dimensional controllers are identified as well. In addition, relations with minimax dynamic games are explored to best assess the importance and generality of the finite-dimensional control systems.