A Note on the Semiglobal Controllability of the Semilinear Wave Equation

We study the internal controllability of the semilinear wave equation vtt(x, t)−∆v(x, t) + f(x, v(x, t)) = 1ωu(x, t) for some nonlinearities f which can produce several non-trivial steady states. One of the usual hypotheses to get global controllability, is to assume that f(x, v)v ≥ 0. In this case, a stabilisation term u = γ(x)vt makes any solution converging to zero. The global controllability then follows from a theorem of local controllability and the time reversibility of the equation. In this paper, the nonlinearity f can be more general, so that the solutions of the damped equation may converge to another equilibrium than 0. To prove global controllability, we study the controllability inside a compact attractor and show that it is possible to travel from one equilibrium point to another by using the heteroclinic orbits.