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Singularities in boundary value problems and exact controllability of hyperbolic systems. (English) Zbl 0778.93007

Optimization, optimal control and partial differential equations. Proc. 1st Fr.-Rom. Conf., Iasi/Rom. 1992, ISNM 107, 77-84 (1992).
Summary: [For the entire collection see Zbl 0752.00054.]
For simplicity we consider control by the Dirichlet data of the solutions of the wave equation. One considers a bounded open subset \(\Omega\) of \(\mathbb{R}^ n\) with boundary \(\Gamma=\partial\Omega\) (assumed to be at least Lipschitz). We denote by \(Q\) the cylinder \(\Omega\times]0,T[\) and by \(\Sigma=\Gamma\times]0,T[\) its lateral boundary. We denote by a mere ‘differentiation in \(t\in]0,T[\) and \(u(0)\) denotes the function \(u\) at time \(t=0\). Our problem is the following. Given an initial position \(u_ 0\) and an initial speed \(u\), on \(\Omega\), we look for a time \(T\) and a “control” \(v\) on \(\Sigma\) such the solution of \[ u''=\Delta u\quad\text{in }Q,\quad u(0)=u_ 0,\;u'(0)=u_ 1,\;u|_ \Sigma=v,\tag{1} \] fulfils \(u(T)=u'(T)=0\). In other words we want to bring the corresponding system to rest at time \(T\) under the action of \(v\). We look for a time \(T\) as small as possible. We also endeavor to minimalize the support \(\Sigma_ 0\) of \(v\).

MSC:

93B05 Controllability

Citations:

Zbl 0752.00054
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