A note on the global controllability of the semilinear wave equation

We study the internal controllability of the semilinear wave equation $$v_{tt}(x,t)-\mathrm{\Delta }v(x,t)+f(x,v(x,t))={\text{Un}}_{\omega }u(x,t)$$ 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 $f(x,v)v\ge 0$. In this case, a stabilisation term $u=\gamma (x)v_t$ 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.