A Maximum Principle for Optimal Control Using Spatially Distributed Pointwise Controllers

The problem of characterizing optimal controls for a class of damped distributed parameter systems is considered. The system is governed by a variable-coefficient, linear-partial differential equation involving up to second-order time derivatives and up to fourth-order space derivatives of the state variable. Pointwise controllers extending over the spatial region occupied by the system are considered. A class of performance indexes is introduced that includes convex functions of the state velocity, spatial state derivatives, and the control force. The state variable and its derivatives are evaluated at a preassigned terminal time. A maximum principle is given that facilitates the determination of the optimal control, which is shown to be unique. The use of the maximum principle is demonstrated by determining the optimal pointwise control of the vibrations for a uniform undamped Euler-Bernoulli beam.