Affiliations: Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail: [email protected] | Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21250 E-mail: [email protected]
Abstract: With observation restricted to a single component: displacement, velocity, or temperature, we consider observability of the nonscalar thermoelastic system [1−γΔ]wtt+Δ2w−αΔθ=0, θt−Δθ+αΔwt=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.
Keywords: parametric asymptotics, minimal energy, blowup, thermoelastic system, observation time, coupling parameter, system components, distributed parameter systems, partial differential equations
