A note on the global controllability of the semilinear wave equation

Romain Joly (1) , Camille Laurent (2)

1 IF - Institut Fourier
2 LJLL - Laboratoire Jacques-Louis Lions

We study the internal controllability of the semilinear wave equation $$v_{tt}(x,t)-\Delta v(x,t) + f(x,v(x,t))= {1\hspace{-1.5mm}1}_{\omega} u(x,t)$$ 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 $f(x,v)v\geq 0$. 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.

Mots clés: global controllability; compact global attractor; internal control; heteroclinic orbits; wave equation

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Romain Joly, Camille Laurent. A note on the global controllability of the semilinear wave equation. SIAM Journal on Control and Optimization, 2014, 52, pp.439-450. ⟨hal-00731117v2⟩