## 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

### Keywords:

Dirichlet data; wave equation

Zbl 0752.00054