The optimal control of linear systems with unknown parameters

The optimal open-loop controller for a linear system with some unknown parameters is determined. It is assumed that the unknown parameters do not vary during the process, and that their probability distribution function is given. The criterion for optimality is a quadratic function of the state and the input whoso expectation with respect to the unknown parameters has to be minimized. An explicit solution for the optimal controller is obtained by using the methods of the calculus of variations. Further, an alternate approach is presented which, for the problem solved, leads to the same optimal controller.