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

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  • 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.
    Mathematics Subject Classification: 35L05, 93B05, 93C20.

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