Solving two-point boundary value problems for a wave equation via the principle of stationary action and optimal control

TL;DR

This paper introduces a novel method for solving two-point boundary value problems for wave equations by leveraging the principle of stationary action within an optimal control framework, enabling explicit solutions via spectral methods.

Contribution

It develops a fundamental solution approach using max-plus convolution and operator differential equations, providing a new computational technique for wave boundary problems.

Findings

Fundamental solution is a quadratic functional derived from operator differential equations.

Solution method is applicable to any boundary data via max-plus convolution.

Spectral methods enable efficient computation of the fundamental solution.

Abstract

A new approach to solving two-point boundary value problems for a wave equation is developed. This new approach exploits the principle of stationary action to reformulate and solve such problems in the framework of optimal control. In particular, an infinite dimensional optimal control problem is posed so that the wave equation dynamics and temporal boundary data of interest are captured via the characteristics of the associated Hamiltonian and choice of terminal payoff respectively. In order to solve this optimal control problem for any such terminal payoff, and hence solve any two-point boundary value problem corresponding to the boundary data encapsulated by that terminal payoff, a fundamental solution to the optimal control problem is constructed. Specifically, the optimal control problem corresponding to any given terminal payoff can be solved via a max-plus convolution of this fundamental solution with the specified terminal payoff. Crucially, the fundamental solution is shown to be a quadratic functional that is defined with respect to the unique solution of a set of operator differential equations, and computable using spectral methods. An example is presented in which this fundamental solution is computed and applied to solve a two-point boundary value problem for the wave equation of interest.