Asymptotic rates of blowup for the minimal energy function for the null controllability of thermoelastic plates: The free case

Continuing the analysis undertaken in References 8 and 9, we consider the nullcontrollability problem for thermoelastic plate partial differential equations (PDEs) models in the absence of rotational inertia, defined on a two-dimensional domain Ω, and subject to the free mechanical boundary conditions of second and third order. It is now known that such uncontrolled systems generate analytic semigroups on finite energy spaces. Consequently, the concept of null controllability is indeed an appropriate question for consideration. It is shown that all finite energy states can be driven to zero by means of L2[(0, T) × Ω] controls in either the mechanical or thermal component. However, the main intent of the paper is to quantify the singularity, as T ↓ 0, of the minimal energy function relative to null controllability. In particular we shall show that in the case of one control function acting upon the system, the singularity of minimal energy is optimal; it is in fact of order O(T− 5/2), which is the same rate of blowup as that of any finite dimensional approximation of the problem. The PDE estimates, which are obtained in the process of deriving this sharp numerology, will have a strong bearing on regularity properties of related stochastic differential equations