A Numerical Method for Controllability Problems for the Wave Equation

Abstract

Let T denote a given positive number and let uo(x) and U _l(x) denote given functions defined on (0,1). Let Σ = {0, 1} × (0, T), Q = (0, 1) × (0, T) and (u ₀, u ₁ ∈ L ²(0, 1) × H ⁻¹(0, 1). The exact Dirichlet boundary controllability problem for the wave equation is: find a control function g(x, t) defined on Σ such that u satisfies

$$ \left\{ {\begin{array}{*{20}{c}} {{{u}_{{tt}}} - {{u}_{{xx}}} = 0\quad in\;Q} \hfill \\ {u{{|}_{{t = 0}}} = {{u}_{0}}\quad and\quad {{u}_{t}}{{|}_{{t = 0}}} = {{u}_{1}}\quad in\;(0,1)} \hfill \\ {u{{|}_{{t = T}}} = 0\quad and\quad {{u}_{t}}{{|}_{{t = T}}} = 0\quad in\;(0,1)} \hfill \\ {u = g\quad on\;\sum .} \hfill \\ \end{array} } \right. $$

((1))