## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 1, Number 1 (1996), 83-102.

### Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping

#### Abstract

In this paper we study a hinged, extensible, and elastic nonlinear beam equation with structural damping and Balakrishnan-Taylor damping with the full exponent $2\left(n+\beta \right)+1$. This strongly nonlinear equation, initially proposed by Balakrishnan and Taylor in 1989, is a very general and useful model for large aerospace structures. In this work, the existence of global solutions and the existence of absorbing sets in the energy space are proved. For this equation, the feature is that the exponential rate of the absorbing property is not a global constant, but which is uniform for the family of trajectories starting from any given bounded set in the state space. Then it is proved that there exists an inertial manifold whose exponentially attracting rate is accordingly non-uniform. Finally, the spillover problem with respect to the stabilization of this equation is solved by constructing a linear state feedback control involving only finitely many modes. The obtained results are robust in regard to the uncertainty of the structural parameters.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 1, Number 1 (1996), 83-102.

**Dates**

First available in Project Euclid: 7 April 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1049725993

**Digital Object Identifier**

doi:10.1155/S1085337596000048

**Mathematical Reviews number (MathSciNet)**

MR1390561

**Zentralblatt MATH identifier**

0940.35036

**Subjects**

Primary: 35B40: Asymptotic behavior of solutions 35L75

Secondary: 73K05 93D15

**Keywords**

Inertial manifold stabilization nonlinear beam equation dissipative solution semigroup Balakrishnan-Taylor damping

#### Citation

You, Yuncheng. Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping. Abstr. Appl. Anal. 1 (1996), no. 1, 83--102. doi:10.1155/S1085337596000048. https://projecteuclid.org/euclid.aaa/1049725993