Localized null controllability and corresponding minimal norm control blowup rates of thermoelastic systems

In this paper, we consider the problem of null controllability for an elastic operator under square root damping. Such partial differential equation models can be described by analytic semigroups on the basic space of finite energy. Thus by inherent smoothing coming from the parabolic-like behavior of the dynamics, the problem of null controllability is appropriate for consideration. In particular, we will show that the solution variables can be steered to the zero state by means of iterations of locally supported steering controls acting on appropriate finite dimensional systems. The hinged boundary conditions considered here admit of a diagonalization of the spatial operator. The control strategy implemented in [A. Benabdallah, M. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal. 7 (2002) 585–599] is used to construct a suboptimal control for the problem, but here we expand upon their results by providing a bound for the energy function Emin(T), T>0. Our results are valid for localized mechanical and thermal control. The strategy relies heavily on the availability of a Carleman's estimate for finite linear combinations of eigenfunctions of the Dirichlet Laplacian.