Exact Controllability of the One-Dimensional Wave Equation with Locally Distributed Control

Abstract

The one-dimensional wave equation with variable wave speed and locally distributed control is considered. It is shown that the adjoint system is observable using a multiplier method with a multiplier being the solution of an ordinary differential equation. It is also shown that a sufficient condition for exact controllability is that a related minimization problem always has an optimal solution. Since the objective function for this minimization problem would be coercive if the adjoint system is observable this establishes the exact controllability of the original system.