Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces.

*(English)*Zbl 0593.34048Let A be the infinitesimal generator of the \(C_ 0\)-semigroup S(t). It is well known that the spectral condition (1) \(\sup \{Re \lambda,\lambda \in \sigma (A)\}<-\delta,\) \(\delta >0\), is not sufficient for S(t) to decay exponentially (to be exponentially stable). See e.g. A. Pazy’s book, p. 117. The author shows that even if we assume in addition to (1), that \((\lambda I-A)^{-1}\) is compact, S(t) may not decay exponentially. Thus he points out that an existing result in this direction is not true. In the last part of the paper he gives necessary and sufficient conditions, in terms of spectral properties of A, for the exponential stability of S(t). Of course, the condition \(\sup \{Re \lambda <- \delta;\lambda \in \sigma (A)\}\) appearing in the statement of Theorems A and B has to be formulated in the standard (and better) form (1) above. We may conclude that the author studies some fundamental relationships between the spectral properties of A and the exponential stability of S(t).

Reviewer: N.H.Pavel

##### MSC:

34D05 | Asymptotic properties of solutions to ordinary differential equations |

35B40 | Asymptotic behavior of solutions to PDEs |

37-XX | Dynamical systems and ergodic theory |

47A50 | Equations and inequalities involving linear operators, with vector unknowns |

34A30 | Linear ordinary differential equations and systems, general |

34G10 | Linear differential equations in abstract spaces |