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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

w ¨+ηw ˙+γA β w=u(t)+f(t,w,u(t)),

on a Hilbert space. The distributed control uL 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.

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