A Numerical Method for Controllability Problems for the Wave Equation

Max D. Gunzburger (), L. Steven Houl () and Lili Ju ()
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Max D. Gunzburger: Iowa State University, Department of Mathematics
L. Steven Houl: Iowa State University, Department of Mathematics
Lili Ju: Iowa State University, Department of Mathematics

A chapter in Hyperbolic Problems: Theory, Numerics, Applications, 2003, pp 557-567 from Springer

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

Date: 2003
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DOI: 10.1007/978-3-642-55711-8_52

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