Abstract
A vibrating plate is here taken to satisfy the model equation:u tt + Δ2u = 0 (whereΔ 2u:= Δ(Δu); Δ = Laplacian) with boundary conditions of the form:u v = 0 and(Δu) v = ϕ = control. Thus, the state is the pair [u, u t] and controllability means existence ofϕ on Σ:= (0,T)×∂Ω transfering ‘any’[u, u t]0 to ‘any’[u, u t]T. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with∥ϕ∥ = O(T −1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation:u tt = Δu) with log∥ϕ∥ = O(T −1) asT→0. Some related results on minimum time control are also included.
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Communicated by I. Lasiecka
This research was partially supported under the grant AFOSR-82-0271.
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Krabs, W., Leugering, G. & Seidman, T.I. On boundary controllability of a vibrating plate. Appl Math Optim 13, 205–229 (1985). https://doi.org/10.1007/BF01442208
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DOI: https://doi.org/10.1007/BF01442208