About global null controllability of a quasi-static thermoelastic contact system

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 [8], 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.