Boundary Value Control of the Wave Equation in a Spherical Region

Abstract

In this paper we study controllability problems for the wave equation \[\frac{{\partial ^2 w}}{{\partial t^2 }} - \sum_{j = 1}^n {\frac{{\partial ^2 w}}{{\left( {\partial x^j } \right)^2 }}} ,\qquad t \geqq 0,\quad x \in \Omega ,\] where $\Omega $ is a spherical region in $R^n $. The control force f enters in the boundary condition \[\frac{{\partial w}}{{\partial v}} = f,\] assumed to hold for $t \geqq 0$, $x \in \Gamma = \partial \Omega $. Our main result is that all “finite energy” initial states can be steered to the zero state in time $\tau $, using a control $f \in L^2 (\Gamma \otimes [0,\tau ])$, provided $\tau > 2$.Beginning with standard existence results for solutions of the wave equation, we show the control problem to be equivalent to a collection of trigonometric moment problems. These are solved using the theory of nonharmonic Fourier series together with certain results concerning the separation of eigenvalues of the Helmholtz operator in $L^2 (\Omega )$ with the Neumann boundary condition. ...