A note on the global controllability of the semilinear wave equation

TL;DR

This paper investigates the internal controllability of the semilinear wave equation with general nonlinearities, demonstrating the ability to steer solutions between different equilibrium states using heteroclinic orbits.

Contribution

It extends controllability results to more general nonlinearities where solutions may converge to various equilibria, not just zero.

Findings

Controllability within a compact attractor is achievable.

Traveling between different equilibria is possible via heteroclinic orbits.

The approach generalizes previous results assuming nonnegative nonlinearities.

Abstract

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