# Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem We study the controllability properties of a nonlinear parabolic system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case the system is proved to be exactly null controllable provided the parameters of the system are smooth. The proof is based on changing the control variable and using Aubin’s Compactness Lemma to obtain an invariant set for the linearized controllability map. Then, by proving that the found solution is sufficiently smooth, we get the null controllability for the original system. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

, Volume 51 (1) – Jan 1, 2005
16 pages

/lp/springer_journal/exact-null-controllability-of-a-nonlinear-thermoelastic-contact-7RvCjiZgU5
Publisher
Springer Journals
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-004-0808-0
Publisher site
See Article on Publisher Site

### Abstract

We study the controllability properties of a nonlinear parabolic system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case the system is proved to be exactly null controllable provided the parameters of the system are smooth. The proof is based on changing the control variable and using Aubin’s Compactness Lemma to obtain an invariant set for the linearized controllability map. Then, by proving that the found solution is sufficiently smooth, we get the null controllability for the original system.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jan 1, 2005

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