Null Controllability of the Von Kármán Thermoelastic Plates under the Clamped or Free Mechanical Boundary Conditions

In this paper, we prove the local exact null controllability of the thermoelastic plate model, in the absence of rotational inertia, and under the influence of the (non-Lipschitz) von Kármán nonlinearity. The plate component may be taken to satisfy either the clamped or higher order (and physically relevant) free boundary conditions. In the accompanying analysis, critical use is made of sharp observability estimates which obtain for the linearization of the thermoelastic plate (these being derived in [3] and [4]). Moreover, another key ingredient in our work to steer the given nonlinear dynamics is the recent result in [10] concerning the sharp regularity of the von Karman nonlinearity. 1 Statement of Problem and Main Results Throughout, Ω ⊂ R will be a bounded, open set with smooth boundary Γ. Given terminal time T , 0 < T < 11, we consider the following nonlinear thermoelastic system on Ω× (0, T ) :  1⁄2 ωtt +∆ ω + α∆θ = [F(ω),ω] θt −∆θ − α∆ωt = u on (0, T )× Ω ω(t = 0) = ω0; ωt(t = 0) = ω1; θ(t = 0) = θ0 on Ω. (1) In this model the parameter α, which couples the hyperbolic-like (plate) and parabolic (heat) dynamics, is nonzero with, say, M ≥ α > 0. Concerning the nonlinearity which appears in the plate component of this system: the so-called von Kármán bracket [·, ·] is defined by having for all v, ṽ ∈ H(Ω), [v, ṽ] = vxxṽyy + vyy ṽxx − 2vxy ṽxy. Moreover, the Airy Stress function F (·) which appears within the bracket in (1) is defined by the solution of the following elliptic problem: ∆F (v) = − [v, v] in Ω; F (v) = ∂F (v) ∂ν ̄̄̄̄