Global attractors for damped abstract nonlinear hyperbolic systems

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Title: Global attractors for damped abstract nonlinear hyperbolic systems
Author: Pinter, Gabriella A.
Abstract: This dissertation is concerned with the long time dynamics of a class of damped abstract hyperbolic systems that arise in the study of certain smart material structures , namely elastomers . The term smart material refers to a material capable of both sensing and responding actively to outside excitation . These properties make smart materials a prime canditate for actuation and sensing in next generation control systems . However , modeling and numerically simulating their behavior poses several difficulties . In this work we consider a model for elastomers developed by H .T . Banks , N .J . Lybeck , B .C . Munoz , L .C . Yanyo , formulate this model as an abstract evolution system , and study the long time behavior of its solutions . We remark that the question of existence and uniqueness of solutions for this class of systems is a challenging problem and was only recently solved by H .T . Banks , D .S . Gilliam and V .I . Shubov . Concerning the long time dynamics of the problem , we first prove that the system generates a weak dynamical system , and possesses a weak global attractor . Our main result is the existence of a "strong" dynamical system which has a compact global attractor . With the help of a Lyapunov function we are able to characterize the structure of this attractor . We also give a theorem that guarantees the stability of the global attractor with respect to varying parameters in the system . Our last result concerns the uniform differentiability of the dynamical system .
URI: http : / /hdl .handle .net /2346 /12123
Date: 1997-05

Citation

Pinter, Gabriella A. Global attractors for damped abstract nonlinear hyperbolic systems. Doctoral dissertation, Texas Tech University. Available electronically from http : / /hdl .handle .net /2346 /12123 .

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