We study the null controllability properties of a 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 at the free end of the rod. We study the control problem
and treat the case when the control is distributed over the whole space domain.
In , we proved that if the initial condition is smooth and the system has a
strong solution, then there is a control that brings the system to zero. The proof was
based on changing the control variable and using Aubin's Compactness Lemma. In
this paper, we focus on the null controllability of the weak solutions. We establish
the existence of a control that steers the system to zero. Our approach consists of
approximating the initial condition by the smooth functions and then proving that
the obtained sequence of strong solutions converges to a weak solution of the desired
type. The uniqueness of a weak solution is established only under special assumptions
on the parameters of the system.