Robustness of strong stability of semigroups
Robustness of strong stability of \(C_0\)-semigroups on Hilbert spaces under additive finite-rank perturbation is studied. Throughout this article, it is assumed that \(A\) generates a strongly stable \(C_0\)-semigroup \((T(t))_{t\geq 0}\) on a Hilbert space \(X\) with \(\sigma(A)\cap i\mathbb R=\{i\omega_k\}_{k=1,\dots,N}\) and the norm of its resolvent operator is polynomially bounded near these points. The first main result of this paper shows that, if \(B\in L(\mathbb C^p,X)\) and \(C\in L(X,\mathbb C^p)\) with \({\mathcal R}(B)\subset {\mathcal R}((i\omega_k-A)^\beta)\) and \({\mathcal R}(C^*)\subset {\mathcal R}((-i\omega_k-A^*)^\beta)\) for all \(k+1,\dots,N\) and some \(\beta+\gamma\) sufficiently large, and the graph norms \(\|B\|+\|(i\omega_k-A)^{-\beta}B\|\) and \(\|C\|+\|(-i\omega_k-A^*)^{-\gamma}C^*\|\) are sufficiently small, then the semigroup generated by \(A+BC\) is strongly stable. The second main result of this paper presents a similar perturbation result for polynomially stable \(C_0\)-semigroups. This improves the perturbation results concerning polynomially stability obtained by the author in [J. Funct. Anal. 263, No. 9, 2555--2583 (2012; Zbl 1263.47049); Syst. Control Lett. 62, No. 4, 331--337 (2013; Zbl 1261.93059)].
scientific article; zbMATH DE number 6359818
scientific article