Asymptotic Properties of a State Density Approximation for Optimal Control

A scalar discrete-time state estimation and stochastic optimal control problem is considered which differs from the standard linear-quadratic-Gaussian case only by the presence in the dynamics of a quadratic term in the state and noise variables with a small coefficient. With respect to this coefficient, precise results are given concerning the asymptotic convergence in probability of a certain approximation for the conditional state probability density function, and of a corresponding approximation for an optimal control law, to first-order asymptotic approximations thereof in the coefficient´s reciprocal. A possible origin of such estimation or control problems from perturbation analyses is described, and the significance of the results in this context is discussed.