On the solution of constrained wave equation problems and an example of constrained energy laws for PDE's

Abstract

This paper has two main purposes. The first is to investigate what happens if we put constraints on the classical wave equation such as limiting the velocity or length of the string. The second purpose is to present a systematic method to solve a wide variety of constrained optimization problems in the calculus of variations which are models of real world problems involving, for example, minimizing the "energy" of a physical process. Of particular interest is the involvement of reasonable inequality constraints with well understood classical problems involving partial differential equations. Because our techniques are new for these problems, much of this paper will be concerned with explaining these techniques.