# American Institute of Mathematical Sciences

July  2003, 9(4): 901-924. doi: 10.3934/dcds.2003.9.901

## Exact controllability in "arbitrarily short time" of the semilinear wave equation

 1 Laboratoire de Mathématiques MIP, UMR CNRS 5640, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse cedex 4, France, France

Received  October 2001 Revised  November 2002 Published  April 2003

We study the exact controllability of the one dimensional semilinear wave equation by a control acting on an open subset $(a,b)$ of the domain $(0,1)$. With the aid of d'Alembert's formula and sidewise energy estimates, we obtain sharp conditions in the space-time support of the control, that coincide with the by now well-known geometric control condition.
More precisely, by classical results of J. Lagnese, A. Haraux and E. Zuazua, exact controllability holds in time $T > T_0$:$= 2 max (a , 1-b)$ and fails if $T < T_0$. We weaken strongly their results: given $T>T_0$, we prove that the control can be chosen so that it is supported only on some special time intervals: they are parts of $(0,T)$, in finite number (depending on $a$ and $b$), and their total length can be arbitrarily small. The only condition is that they have to be "close enough" from each other. If this condition holds, we study the observability cost. If it fails, we prove that exact controllability in time $T$ does not hold, but can still be true in time $T'$ large enough.
Citation: Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901
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