Two-point boundary value problems and exact controllability for several kinds of linear and nonlinear wave equations

TL;DR

This paper introduces new concepts for second-order hyperbolic equations, proving existence of solutions and establishing global exact controllability for various linear and nonlinear wave equations, with applications to geometric flows.

Contribution

It develops the theory of two-point boundary value problems and proves exact controllability for several wave equations, including those on closed curves and strips, with applications to hyperbolic curvature flow.

Findings

Existence of smooth solutions for two-point boundary value problems.

Global exact controllability of several wave equations.

Application to hyperbolic curvature flow and geometric analysis.

Abstract

In this paper we introduce some new concepts for second-order hyperbolic equations: two-point boundary value problem, global exact controllability and exact controllability. For several kinds of important linear and nonlinear wave equations arising from physics and geometry, we prove the existence of smooth solutions of the two-point boundary value problems and show the global exact controllability of these wave equations. In particular, we investigate the two-point boundary value problem for one-dimensional wave equation defined on a closed curve and prove the existence of smooth solution which implies the exact controllability of this kind of wave equation. Furthermore, based on this, we study the two-point boundary value problems for the wave equation defined on a strip with Dirichlet or Neumann boundary conditions and show that the equation still possesses the exact controllability in these cases. Finally, as an application, we introduce the hyperbolic curvature flow and obtain a result analogous to the well-known theorem of Gage and Hamilton for the curvature flow of plane curves.