Nonuniformly Attracting Inertial Manifolds and Stabilization of Large Space Structures
by
Yuncheng You
University of South Florida

Consider the long-time dynamics and the stabilization of a nonlinear large space structure whose model equation is

\frac\partial2u\partialt2+\alphaAu+\deltaA\frac12\frac\partialu\partialt+[a+b|A\frac14u|2+q á A\frac12u, \frac\partialu\partialt ñ2(n+\beta)+1]A\frac12u=f,

where the parameters \alpha, \delta, b, and q are positive constants, but a in R. This general model represents beam equations and plate equations as typical large space structures that have structural damping and Balakrishnan-Taylor damping with exponents n (integer >= 0) and 0 <= \beta < 1/2. The original motivation to study this equation was the spillover problem: whether one can design a feedback control f only involving finitely many modes and achieving high performance such as a robust stabilization with a uniform rate.

In this paper, we first prove the existence of global solutions and the dissipativity in the energy space of the semiflow generated by the uncontrolled equation. Then it is proved that there exists an inertial manifold whose exponential attracting rate is however nonuniform. This result implies the finite-dimensional characterization of the long-time dynamics of the uncontrolled vibrations of such large space structures.

Finally, based on the inertial manifold results, the spillover problem associated with stabilizability of this equation is constructively solved by a linear feedback control that involves only finite modes. Moreover, the stabilization as well as inertial manifolds are robust with respect to the uncertainty in the structural parameters.

Date received: December 29, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadd-06.