A minimum principle for smooth first-order distributed systems

The problem of characterizing optimal controls for a class of distributed parameter systems is considered. The system dynamics are characterized mathematically by a finite number of coupled partial differential equations involving first-order time and space derivatives of the state variables. Boundary conditions on the state are in the form of a finite number of algebraic relations between the state and boundary control variables. Multiple distributed controls extending over the entire spatial region occupied by the system are also included. The performance index is an integral over the spatial domain of penalty functions on the terminal state and on the distributed state and controls. Under certain differentiability and well-posedness assumptions, variational methods are used to derive first- and second-order necessary conditions for a control which minimizes the performance index. Of particular interest are conditions on the boundary value of the costate and on the optimal boundary controls.