Hybrid solution method for dynamic programming equations for MDOF stochastic systems

An optimal control problem is considered for a multi-degree-of-freedom (MDO F) system, excited by a white-noise random force. The problem is to minimiz e the expected response energy by a given time instant T by applying a vect or control force with given bounds on magnitudes of its components. This pr oblem is governed by the Hamilton-Jacobi-Bellman, or HJB, partial different ial equation. This equation has been studied previously [1] for the case of a single-degree-of-freedom system by developing a "hybrid" solution. Speci fically, an exact analitycal solution has been obtained within a certain ou ter domain of the phase plane, which provides necessary boundary conditions for numerical solution within a bounded in velocity inner domain, thereby alleviating problem of numerical analysis for an unbounded domain. This hyb rid approach is extended here to MDOF systems using common transformation t o modal coordinates. The multidimensional HJB equation is solved explicitly for the corresponding outer domain, thereby reducing the problem to a set of numerical solutions within bounded inner domains. Thus, the problem of b ounded optimal control is solved completely as long as the necessary modal control forces can be implemented in the actuators. If, however, the contro l forces can be applied to the original generalized coordinates only, the r esulting optimal control law may become unfeasible. The reason is the nonli nearity in maximization operation for modal control forces, which may lead to violation of some constraints after inverse transformation to original c oordinates. A semioptimal control law is illustrated for this case, based o n projecting boundary points of the domain of the admissible transformed co ntrol forces onto boundaries of the domain of the original control forces. Case of a single control force is considered also, and similar solution to the HJB equation is derived.