Appl Math Optim 51:107–122 (2005)
2004 Springer Science+Business Media, Inc.
Exact Null Controllability of a Nonlinear
Thermoelastic Contact Problem
Irina F. Sivergina
and Michael P. Polis
Institute of Mathematics and Mechanics, Russian Academy of Science,
Department of Science and Mathematics, Kettering University,
Flint, MI 48504, USA
School of Engineering and Computer Science, Oakland University,
Rochester, MI 48309, USA
Communicated by I. Lasiecka
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 Com-
pactness 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.
Key Words. Null controllability, Thermoelastic system, Quasi-static approxima-
tion, Nonlocal parabolic equation, Nonlinear boundary conditions.
AMS Classiﬁcation. 93B05, 49J20, 74B05.