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Boundary controllability in transmission problems for thin plates. (English) Zbl 0801.93012

Elworthy, K.D. (ed.) et al., Differential equations, dynamical systems, and control science. A Festschrift in Honor of Lawrence Markus. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 152, 641-658 (1994).
The author considers exact controllability, using boundary controls, for a thin elastic plate having discontinuous, but piecewise constant elastic parameters. “Elastic plate” stands for the Reissner-Mindlin plate model. Let \(\Omega\) represent the region occupied by mid-plane of the plate in equilibrium, \(\Omega_ 1\) an open set, \(\overline {\Omega}_ 1\subset \Omega\), \(\Omega_ 2= \Omega\setminus \Omega_ 1\), \(\Gamma_ 1= \partial\Omega_ 1\), \(\Gamma_ 2= \partial\Omega_ 2\), \(\Gamma= \partial\Omega\), \(P_ 1= \Omega_ 1\times (-{h\over 2}, +{h\over 2})\), \(P_ 2= \Omega_ 2\times (- {h\over 2}, +{h\over 2})\). This lengthy paper consists primarily of proofs of two basic theorems. The first theorem proves the well-posedness of the control problem. The second theorem states that for some \(T_ 0>0\) (there exists such \(T_ 0\)!) for every \(T>T_ 0\) the reachable set \(R_ T= \{\Phi, \dot\Phi\}\), will comprise the entire Hilbert space ‘\(V\times H\)’. Here \(\Phi\) denotes a unique solution of the system corresponding to some admissible control \(F\in L^ 2 ([0,T], U)\).
For the entire collection see [Zbl 0780.00045].
Reviewer: V.Komkov (Roswell)

MSC:

93B05 Controllability
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