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Uniform stability of displacement coupled second-order equations. (English) Zbl 0977.34063
The author considers the system \[ {\partial_{tt} w\choose \partial_{tt} z}+S {w\choose z}+{0\choose D\partial_tz} =0, \] where \(S\) is an operator on \(H\times G\), \(H\) and \(G\) are Hilbert spaces and \(D\) is a positive, selfadjoint operator on \(G\). The goal is to find conditions which guarantee that the energy of the system has an exponential decay to zero, i.e., the system is uniformly stable. This problem was motivated by comments in [D. L. Russell, J. Math. Anal. Appl., 173, No. 2, 339-358 (1993; Zbl 0771.73045)]. The author’s result assumes that \(S\) is representable in operator matrix form as \(S=\left[ \begin{smallmatrix} A & B\\ B^* & C\end{smallmatrix} \right]\), \(A,C\) are positive, selfadjoint, there exists \(c\in [0,{1\over 2})\) such that \(|(u,Bv)_H |< c(\|A^{1\over 2}u \|^2_H+ \|C^{1\over 2}v \|^2_G)\) (which implies that \(S\) is a positive selfadjoint operator), \(B_1 \equiv A^{-{1\over 2}}B\) is boundedly invertible and the operators \(D^{-{1\over 2}} B_1^{-1}A^{1\over 2}\), \(D^{-{1\over 2}} C_1B_1^{-1} A^{-{1\over 2}}\), \(D^{1\over 2}B_1^{-1}\) and \(A^{-{1\over 2}} B_1^{-1}D^{-{1 \over 2}}\) all extend to bounded operators on \(H\times G\), with \(C_1\equiv C-B^*A^{-1}B\). Applications of these ideas are given in the last two sections.

MSC:
34H05 Control problems involving ordinary differential equations
93B05 Controllability
34G20 Nonlinear differential equations in abstract spaces
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
35B37 PDE in connection with control problems (MSC2000)
37N35 Dynamical systems in control
93B52 Feedback control
93D15 Stabilization of systems by feedback
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