Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations

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September 2006, 5(3): 639-658. doi: 10.3934/cpaa.2006.5.639

Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations

Enrique Fernández-Cara ^1, , Manuel González-Burgos ^2, and Luz de Teresa ^3,

1.	Dpto., E.D.A.N., Universidad de Sevilla, Aptdo. 1180; 41080 Sevilla
2.	Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
3.	Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico

Received April 2005 Revised April 2006 Published June 2006

Abstract
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This paper is concerned with the null-exact controllability of a cascade system formed by a semilinear heat and a semilinear wave equation in a cylinder $\Omega \times (0,T)$. More precisely, we intend to drive the solution of the heat equation (resp. the wave equation) exactly to zero (resp. exactly to a prescribed but arbitrary final state). The control acts only on the heat equation and is supported by a set of the form $\omega \times (0,T)$, where $\omega \subset \Omega$. In the wave equation, the restriction of the solution to the heat equation to another set $\mathcal O \times (0,T)$ appears. The nonlinear terms are assumed to be globally Lipschitz-continuous. In the main result in this paper, we show that, under appropriate assumptions on $T$, $\omega$ and $\mathcal O$, the equations are simultaneously controllable.

Keywords: observability inequalities., Semilinear systems, parabolic-hyperbolic equations, controllability.

Mathematics Subject Classification: 35M20, 93B05, 93B0.

Citation: Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639

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