Blowup estimates for observability of a thermoelastic system

Abstract

With observation restricted to a single component: displacement, velocity, or temperature, we consider observability of the nonscalar thermoelastic system [1−γΔ]w_tt+Δ²w−αΔθ=0,  θ_t−Δθ+αΔw_t=0, coupling heat conduction with a Kirkhoff or Euler–Bernoulli plate model. One does have observability in arbitrarily short time here, but necessarily has blowup of the sensitivity as the observation time T→0 and also as the coupling coefficient α→0. In this paper we are able to examine the asymptotics of this blowup for two situations: global observation (i.e., on all of Ω) and, with significant restrictions, boundary observation. The blowup rates obtained are of optimal order: 𝒪(T^−5/2) for global observation, corresponding to what is known for 3-dimensional systems, and exponential in 𝒪(1/T) for boundary observation, corresponding to what is known for scalar PDE problems. Our methods permit us also to obtain asymptotics as α→0 – a question which can only arise for systems.