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The paper is concerned with a dissipativity theory and robust performanceanalysis of discrete-time stochastic systems driven by a statisticallyuncertain random noise. The uncertainty is quantified by the conditionalrelative entropy of the actual probability law of the noise with respect to anominal product measure corresponding to a white noise sequence. We discuss abalance equation, dissipation inequality and superadditivity property for thecorresponding conditional relative entropy supply as a function of time. Theproblem of minimizing the supply required to drive the system between givenstate distributions over a specified time horizon is considered. Suchvariational problems, involving entropy and probabilistic boundary conditions,are known in the literature as Schroedinger bridge problems. In application tocontrol systems, this minimum required conditional relative entropy supplycharacterizes the robustness of the system with respect to an uncertain noise.We obtain a dynamic programming Bellman equation for the minimum requiredconditional relative entropy supply and establish a Markov property of theworst-case noise with respect to the state of the system. For multivariablelinear systems with a Gaussian white noise sequence as the nominal noise modeland Gaussian initial and terminal state distributions, the minimum requiredsupply is obtained using an algebraic Riccati equation which admits aclosed-form solution. We propose a computable robustness index for such systemsin the framework of an entropy theoretic formulation of uncertainty and providean example to illustrate this approach.

State distributions and minimum relative entropy noise sequences in uncertain stochastic systems: the discrete time case