Global smooth solutions and stabilization of nonlinear elastodynamic systems with locally distributed dissipation
Zhi-Fei Zhanga, b, , and Peng-Fei Yaob
aSchool of Mathematics, Wuhan University, Wuhan, Hubei, 430072, PR China
bKey Laboratory of Control and Systems, Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, PR China
Received 27 June 2008;
revised 12 February 2009;
accepted 12 February 2009.
Available online 9 March 2009.
We consider the existence of global solutions of the nonlinear elastodynamic system with a locally distributed damping in a bounded domain. We assume that the energy function satisfies the strong ellipticity condition at the zero equilibrium. The local dissipation is in the form where the nonnegative function a(x) is only positive on a small portion of the domain. We show the existence of global smooth solutions when initial data are small. In particular, we obtain the exponential decay of the energy, which implies the exponential stabilization of the system by internal feedback.
Keywords: Nonlinear elastodynamic systems; Energy estimate; Locally distributed damping