"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.