  The paper is concerned with a dissipativity theory and robust performance
analysis of discrete-time stochastic systems driven by a statistically
uncertain random noise. The uncertainty is quantified by the conditional
relative entropy of the actual probability law of the noise with respect to a
nominal product measure corresponding to a white noise sequence. We discuss a
balance equation, dissipation inequality and superadditivity property for the
corresponding conditional relative entropy supply as a function of time. The
problem of minimizing the supply required to drive the system between given
state distributions over a specified time horizon is considered. Such
variational problems, involving entropy and probabilistic boundary conditions,
are known in the literature as Schroedinger bridge problems. In application to
control systems, this minimum required conditional relative entropy supply
characterizes the robustness of the system with respect to an uncertain noise.
We obtain a dynamic programming Bellman equation for the minimum required
conditional relative entropy supply and establish a Markov property of the
worst-case noise with respect to the state of the system. For multivariable
linear systems with a Gaussian white noise sequence as the nominal noise model
and Gaussian initial and terminal state distributions, the minimum required
supply is obtained using an algebraic Riccati equation which admits a
closed-form solution. We propose a computable robustness index for such systems
in the framework of an entropy theoretic formulation of uncertainty and provide
an example to illustrate this approach.
