On optimal steering of a non-Markovian Gaussian process

At present, the problem to steer general non-Markovian processes between specified end-point marginal distributions with minimum energy remains unsolved. Herein, we consider the special case of a non-Markovian process y(t) which assumes a finite-dimensional stochastic realization with a Markov state process that is fully observable. In this setting, and over a finite time horizon [O, T], we determine an optimal (least) finite-energy control law that steers the stochastic system to a final distribution that is compatible with a specified distribution for the terminal output process y(T); the solution is given in closed-form. This work provides a key step towards the important problem to steer a stochastic system based on partial observations of the state (i.e., an output process) corrupted by noise.