A Class of Variational Problems in Search Theory and the Maximum Principle

Abstract

In this paper we consider certain nonlinear and constrained variational problems that occur in the theory of search. The problems are definitely of a nonclassical variety, but, by using a device of R. Courant, a sequence of unconstrained variational problems is generated, the solutions of which converge to the solution of the constrained problem. The maximum principle of L. Pontryagin et al., is then utilized to construct a corresponding sequence of nonlinear, two-point boundary value problems such that a solution to the variational problem will then necessarily satisfy the boundary value problem. An effective computational method, based on the notion of a generalized Newton-Raphson operator, is then applied to solve iteratively, in a rapid and accurate manner, the nonlinear boundary value problem; under suitable assumptions, this method yields solutions to the variational problem. The technique is illustrated in several examples.