Optimal Steering of a Linear Stochastic System to a Final Probability Distribution - Part III

The subject of this work has its roots in the so-called Schrödginer bridge problem (SBP) which asks for the most likely distribution of Brownian particles in their passage between observed empirical marginal distributions at two distinct points in time. Renewed interest in this problem was sparked by a reformulation in the language of stochastic control. In earlier works, presented as Part I and Part II, we explored a generalization of the original SBP that amounts to optimal steering of linear stochastic dynamical systems between state-distributions, at two points in time, under full state feedback. In these works, the cost was quadratic in the control input, i.e., control energy. The purpose of the present work is to detail the technical steps in extending the framework to the case where a quadratic cost in the state is also present. Thus, the main contribution is to derive the optimal control in this case which in fact is given in closed-form (Theorem 1). In the zero-noise limit, we also obtain the solution of a (deterministic) mass transport problem with general quadratic cost.