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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 
More precisely, by classical results of J. Lagnese, A. Haraux and E. Zuazua, exact controllability holds in time $T > T_0 $:$= 2 max (a , 1b)$ 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.
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