Stochastic uncertain systems subject to relative entropy constraints: Induced norms and monotonicity properties of minimax games

Entropy and relative entropy are fundamental concepts on which information theory is founded on, and in general, telecommunication systems design. On the other hand, dissipation inequalities, minimax strategies, and induced norms are the basic concepts on which robustness of uncertain control and estimation of systems are founded on. In this paper, the precise relation between these notions is investigated. In particular, it will be shown that the higher the dissipation the higher the entropy of the system, which has implications in computing the induced norm associated with robustness. These connections are obtained by considering stochastic optimal uncertain control systems, in which uncertainty is described by a relative entropy constraint between the nominal and uncertain measures, while the pay-off is a linear functional of the uncertain measure. This is a minimax game, in which the controller measure seeks to minimize the pay-off, while the disturbance measure aims at maximizing the pay-off. Salient properties of the minimax solution are derived, including a characterization of the optimal sensitivity reduction, computation of the induced norm, monotonicity properties of minimax solution, and relations between dissipation and relative entropy of the system. The theory is developed in an abstract setting and then applied to nonlinear partially observable continuous-time uncertain controlled systems, in which the nominal and uncertain systems are described by conditional distributions. In addition, existence of the optimal control policy among the class of policies known as wide-sense control laws is shown, and an explicit formulae for the worst case conditional measure is derived. The results are applied to linear-quadratic-Gaussian problems.