Minimum Relative Entropy State Transitions in Linear Stochastic Systems: the Continuous Time Case

This paper is concerned with a dissipativity theory for dynamical systems governed by linear Ito stochastic differential equations driven by random noise with an uncertain drift. The deviation of the noise from a standard Wiener process in the nominal model is quantified by relative entropy. We discuss a dissipation inequality for the noise relative entropy supply. The problem of minimizing the supply required to drive the system between given Gaussian state distributions over a specified time horizon is considered. This problem, known in the literature as the Schroedinger bridge, was treated previously in the context of reciprocal processes. A closed-form smooth solution is obtained for a Hamilton-Jacobi equation for the minimum required relative entropy supply by using nonlinear algebraic techniques.