Stochastic control of bilinear systems: The optimal quadratic controller

For a bilinear stochastic system described by Ito equations, the following problem is considered: find the optimal feedback control law in a class of quadratic controllers. The optimality criterion is the classical quadratic one for a fixed-interval state-regulation problem. It will be shown that the solution is a linear map of optimal-quadratic state estimate. The latter is obtained by adapting to the present feedback-control case the quadratic filter for bilinear systems, which is available in literature for the case of an open-loop system. The matrix function that solves the suboptimal quadratic-feedback control problem results in the same one of the suboptimal linear-feedback case.