A Study of Numerical Methods for Boundary Control of the Wave Equation in 1D

Abstract

Consider the linear wave equation in 1D: $$ \begin{gathered} u'' - {u_{{xx}}} = 0,\quad (t,x) \in (0,T)x(0,1), \hfill \\ u(t,0) = 0,\quad u(t,1) = v(t),\quad t \in (0,T), \hfill \\ u(0, \cdot ) = {u^0},\quad u'(0, \cdot ) = {u^1}, \hfill \\ \end{gathered} $$ (1) where′ and x denote time and space derivatives respectively, T > 0, v ∈ L 2(0, T) and (u 0, u 1) ∈ H 0 1 (0,1) × L 2(0,1). The goal is now to find v(t) such that u(T, ·) = u′(T, ·) = 0. If this is possible for all initial data, the system is called exact controllable. The system (1) is exact controllable if and only if T ≥ T 0 = 2.