Certain nonlinear partially observable stochastic optimal control problems with explicit control laws equivalent to LEQG/LQG problems

This paper is concerned with partially observed stochastic optimal control problems when nonlinearities enter the dynamics of the unobservable state and the observations as gradients of potential functions. Explicit representations for the information state are derived in terms of a finite number of sufficient statistics. Consequently, the partially observed problem is recast as one of complete information with a new state generated by a modified version of the Kalman filter. When the terminal cost is quadratic in the unobservable state and includes the integral of the nonlinearities, the optimal control laws are explicitly computed, similar to linear-exponential-quadratic-Gaussian (LEQG) and linear-quadratic-Gaussian (LQG) tracking problems. The results are applicable to filtering and control of Hamiltonian systems.