Minimum Relative Entropy State Transitions In Discrete Time Systems With Statistically Uncertain Noise

We develop a stochastic dissipativity theory for discrete-time systems driven by an uncertain random noise. The deviation of the unknown probability law of the noise from a nominal white noise distribution is quantified by the conditional relative entropy given the initial state of the system. We establish a dissipation inequality and superadditivity property for the conditional relative entropy supply. The problem of minimizing the supply required to drive the system between given state distributions over a specified time horizon is considered. We obtain a dynamic programming Bellman equation for the minimum required relative entropy supply and show that the optimal noise is Markov with respect to the state of the system. For linear systems with Gaussian nominal noise and Gaussian initial and terminal state distributions, computing the minimum required supply is reduced to solving an algebraic Riccati equation.