Feedback bounded control of oscillation for wave equation under a white-noise excitation

Abstract

An optimal bounded feedback control of oscillation problem for wave equation under a white noise random excitation is considered. The purpose of optimal problem is to minimize expected response energy at given time instant (Mayer problem) or integral of magnitude for response energy (Lagrange problem). The bounded control forces are placement at the fixed points (actuators) or distributed in the system. Using the decomposition method this problem can be reduced to the optimal problem for the system of infinite number stochastic differential equation. Taking into account only finite part of this system and using dynamic programming method for expected energy of the system leads to the Cauchy problem for Hamilton-Jacobi-Bellman equation in unbounded domain. The solution of this equation provides both optimal control low and values of the relevant function. The basic difficulty with this approach is searching the solution for multidimensional nonlinear PDE within an unbounded domain. Specially, an exact analytical solution has been obtained within a certain unbounded outer domain on the phase plane, which did not contain include switching line of the control. As a justification of this approach it is proved that correspondent outer solution is approximate solution for HJB equation in outing domain. The values of Bellman function H at the boundary of outer domain provide necessary boundary conditions for numerical solution within bounded (finite) inner domain, which is a complement of outer domain. The size of outer domain can be chosen such a way that the values of Bellman function and its corresponding derivatives will coincide at the boundary of outer and inner domain with the corresponding values obtained as a result of numerical solution within inner domain. The analytical outer solution together with the numerical inner solution complete hybrid solution for HJB equation, which make it possible to find out the optimal control low. As an example the control problems for beam and plate are considered.