Optimal control in linear partially observable systems with unknown parameters

The paper is devoted to the investigation of optimal control in a linear time-dependent system with a quadratic functional. Both system coefficients and the performance index depend on an unknown random parameter. At the initial moment, the parameter belongs to a finite set with a probability measure on it. In the course of control process, information refinement regarding the parameter takes place at discrete time moments. This refinement consists of deleting extreme values in the parameter set and of a redefinition of the probabilistic distribution measure. An algorithm for constructing optimal control in a linear-quadratic optimization problem with multiplicative noise is developed. It is shown that for an arbitrary distribution of the unknown parameter, the optimal control can be found in a closed analytic form. The suggested algorithm has a sequential nature: after receiving more accurate information, the observer determines the 'full' optimal control for the whole remaining time interval and applies its 'left tail' until the next observation only. After receiving new information, the performance index is redefined, and the control is recalculated. A problem of optimal control for a singularly perturbed system with incomplete information is considered. Small parameter at the derivative appears when the characteristic damping time for the transition process in the system is much smaller than the interval between observations. For these systems, 'myopic' control policies are constructed and are obtained by minimizing the performance index over the forthcoming observation interval. Calculation of myopic policies is computationally simpler than computation of full policies. It is shown that the myopic policies are asymptotically accurate. An example demonstrating computations and a comparison of full and myopic policies is presented.