2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.
Electron. J. Diff. Eqns., Conference 13, 2005, pp. 75-88.

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

Hugo Leiva

Abstract:
In this paper, 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.

Published May 30, 2005.
Math Subject Classifications: 34G10, 35B40.
Key Words: Non-linear generalized wave equations; strongly continuous groups; exact controllability; Sine-Gordon equation.

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Hugo Leiva
Department of Mathematics
Universidad de los Andes
Merida 5101, Venezuela
email: hleiva@ula.ve

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