1 Stochastic Uncertain Systems Subject to Relative Entropy Constraints: Induced Norms and Monotonicity Properties of Minimax Games Charalambos D. Charalambous and Farzad Rezaei Abstract— Entropy and relative entropy are fundamental con- cepts 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 con- trolled 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. Index Terms— Uncertain Stochastic Systems, Large Deviations, Relative Entropy, Minimax Games, Duality Properties. I. I NTRODUCTION This paper is concerned with non-parametric robust signal processing, estimation and control techniques, in which the un- certain description of the system, and the nominal description of the system are modeled by probability distributions, or gen- eral measures, defined on measurable spaces. These measures are quite general; they may correspond to memoryless random processes as well as random processes, which are solutions of linear as well as nonlinear stochastic dynamical systems. The This work is supported by the European Commission under the Marie Curie grant ICCCSYSTEMS and by a Medium Size University of Cyprus grant. C.D. Charalambous is with the Department of Electrical and Computer Engineering, University of Cyprus, 75 Kallipoleos Avenue, Nicosia, CYPRUS. Also an Adjunct Professor with the School of Information Technology and Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario, K1N 6N5, CANADA. F. Rezaei is a Ph.D. student with the School of Information Technology and Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario, K1N 6N5, CANADA. uncertainty description of these systems is characterized by the class of uncertain measures which satisfy a relative entropy constraint, with respect to the nominal measure. The problem of robust estimation or control is formulated by minimizing over the set of controls or estimators, the maximum of a linear functional of the uncertain measure over the relative entropy constraint set. In the parlame of robustness, this approach leads to minimax techniques, in which the worst case estimate of the uncertain measure subject to the uncertainty description is sought. The theory and contributions of this paper are developed at two levels of generality; the abstract level and the application level. At the abstract level, a general framework is put forward in which the basic ideas are explained, and the fundamental results are derived. At this level, systems are represented by measures on measurable spaces, energy signals by linear functionals on the space of measures, and uncertainty by sets described by bounded relative entropy between the true mea- sure and the nominal measure. The objective of the abstract level formulation is to seek answers to the following questions. • Is the induced norm of the H ∞ criterion (or optimal sensitivity reduction) related to the abstract formulation of the minimax game, and if so, how is it computed? • Is there a relation between dissipation and entropy of the system? If there is a relation between dissipation and entropy, how does it relate to the Second Law of Thermodynamics (which states that higher dissipation implies higher entropy)? • What type of monotonicity properties do the minimax strategies enjoy in the dual space? Specifically, is the Free Energy and Relative Entropy monotonic with respect to the Langrange multiplier associated with the Relative Entropy constraint? • Is there a lower and upper bound on the worst case performance expressed in terms of a priori information of the system, such as, the nominal measure? Answers to these questions are obtained and articulated in the first part of the paper. At the application level, the theory is applied to complex systems, with emphasis on nonlinear partially observable stochastic control systems described by stochastic differential equations. The main objective here, is to transform this class of problems into equivalent problems so that the results and conclusions obtained at the abstract level are easily applicable. Additional issues which are addressed here are, the computa-