Optimal control of a class of nonlinear distributed parameter systems

The objective of this dissertation is to develop optimal control theory and to formulate a practical design method for a class of systems described by nonlinear partial differential equations (PDE) with a general cost functional. System analysis and control design are carried out based on the complete PDE model. The resulting infinite-dimensional optimal control is used as a reference to facilitate implementable finite-dimensional control. Toward the practical applications, we have introduced a new definition of admissible control, derived an integral version of maximum principle using dynamic programming and Sobolev space Frechet differential theory, and presented the costate equation in a general tensor notation. As special cases, we have studied the optimal control and optimal state estimation problems for quantum mechanical systems. We have studied the optimal regulation problem for a two-dimensional hyperbolic system. A general structure for the feedback control has been found. It has been shown that the linear component (in terms of system state) of the optimal feedback control is equal to the solution of the LQ problem for the linearized system. We have developed a target approximation method to facilitate a low dimension controller. This method uses a robust infinite dimensional closed-loop system as a reference. It designs a feedback controller and the actuator and sensor locations in such a way that the resulting feedback system imitates the target so that the closed-loop stability and design performance are preserved. We have studied in detail a nonlinear string vibration suppression problem. All the control design techniques developed in this dissertation are demonstrated on this example.