Feedback control for a heat equation under a white-noise excitation

An optimal feedback control problem for a heat conduction equation under white-noise random excitation is considered. The problem is to minimize expected response for integral value of squared difference among current and preassigned temperature during a given time instant T. The control forces are concentrated in the fixed points (heat actuators). The magnitude of control forces are restricted by positive values. Using dynamic programming method this problem can be reduced to the Couchy problem for Hamilton-Jacobi-Bellman (HJB) partial nonlinear differential equation for a Bellman function H in unbounded domain. Specifically, an exact analytical solution has been obtained within a certain unbounded outer domain on the phase plane, which provides necessary boundary conditions for numerical solution within a bounded (finite) inner domain, thereby alleviating problem of numerical analysis for an unbounded domain. The size of outer domain can be chosen such way that the values of Bellman function H and its corresponding derivatives will coincide in the boundary of outer and inner domains. As an example the case of control problem for one heat actuator in the rod is considered.