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.