"2003 Colloquium on Differential Equations and Applications,
 Maracaibo, Venezuela.
Electronic Journal of Differential Equations,
Conference 13, 2005, pp. 75-88.

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

Author: Hugo Leiva (Univ. de los Andes, Merida, Venezuela)
 
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.