Uniform stability of displacement coupled second-order equations (Q2718941)

The author considers the system NEWLINE\[NEWLINE{\partial_{tt} w\choose \partial_{tt} z}+S {w\choose z}+{0\choose D\partial_tz} =0,NEWLINE\]NEWLINE 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 [\textit{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.