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Exact controllability of a non-linear generalized damped wave equation: Application to the sine-Gordon equation. (English) Zbl 1144.34354
Summary: We give a sufficient conditions for the exact controllability of the non-linear generalized damped wave equation $$ \ddot{w}+ \eta \dot{w} + \gamma A^{\beta} w = u(t) + f(t, w, u(t)), $$ 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.

34G10Linear ODE in abstract spaces
35B40Asymptotic behavior of solutions of PDE
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