A Non-Linear Optimum Stochastic Control Problem

摘要

Optimum control of a vector-valued dynamic system with white noise driving functions is considered. Observations are taken to be linear in the system vector and in the observation error process which is also a white noise. The convariance matrices of the white noise processes and the coefficients of the dynamic system and of the observation equation are all assumed to be known. The loss function is a linear combination of the square of a one-dimensional final miss distance and the integral over time of the absolute value of the control vector. The optimum control is to be determined as a function of all available observations. By attacking the discrete control problem (control is restricted to a finite number of previously specified times) it is shown that the optimum control is a function only of the 'best' estimate of the final miss and that control is to be applied in such a way that this estimate remains within a certain boundary. Thus, the optimum control problem is reduced to that of determining this boundary. For the discrete case, recursion equations are given by which the boundary can be computed. By formally passing to the continuous case, under certain regularity conditions it is shown that this boundary is the solution to a free boundary problem for the heat equation. (Author)