Problems of stationarity and control with a nonlocal cost per unit time

Abstract

The classical calculus of variations is extended in such a fashion as to account for Lagrangian functions that contain a finite number of functionals as arguments. These results are applied to two classes of problems in which the cost per unit time depends on the total allocation over a fixed interval, and the total allocation is linear functional of the process. All appropriate initial data and the total allocation are determined for problems in which the cost functional is to be stationarized such that the process starts and ends at specified times with specified values. The first class of problems considers only the simple stationarization. The second class of problems adjoins a law of evolution and control to the problems of the first class and seeks the stationarizing allocation in addition to the information required in the first class of problems. The effects of nonlinearity are pointed out, and it is shown that a singularity of the algebraic problem associated with the boundary conditions always results in a unique solution while the nonsingular case admits both the possibility of no solutions or of many solutions.