Exact controllability of a non-linear generalized damped wave equation: Application to the sine-Gordon equation.

*(English)*Zbl 1144.34354Summary: We give a sufficient conditions for the exact controllability of the non-linear generalized damped wave equation

$$\ddot{w}+\eta \dot{w}+\gamma {A}^{\beta}w=u\left(t\right)+f(t,w,u\left(t\right)),$$

on a Hilbert space. The distributed control $u\in {L}^{2}$ and the operator $A$ is positive definite self-adjoint unbounded with compact resolvent. The non-linear term $f$ is a continuous function on $t$ and globally Lipschitz in the other variables. We prove that the linear system and the non-linear system are both exactly controllable; that is to say, the controllability of the linear system is preserved under the non-linear perturbation $f$ . As an application of this result one can prove the exact controllability of the sine-Gordon equation.