New characterizations of asymptotic stability for evolution families on Banach spaces.

*(English)*Zbl 1071.47044Uniform exponential stability of operator semigroups and evolution semigroups has been investigated intensely in the last decades. The importance of these questions lies in the investigation of the asymptotic behaviour of infinite-dimensional differential equations.

In the present paper, the authors generalize known results on the uniform exponential stabiluty of operator semigroups by Datko, Pazy, van Neerven and others to evolution families. The main idea of their proof is to use the well-etablished concept of evolution semigroups. They obtain various characterizations of uniform exponential stability, as well as sufficient conditions and finally an individual stability result.

Although mathematically interesting, unfortunately the paper is not carefully written and contains many typographical errors.

In the present paper, the authors generalize known results on the uniform exponential stabiluty of operator semigroups by Datko, Pazy, van Neerven and others to evolution families. The main idea of their proof is to use the well-etablished concept of evolution semigroups. They obtain various characterizations of uniform exponential stability, as well as sufficient conditions and finally an individual stability result.

Although mathematically interesting, unfortunately the paper is not carefully written and contains many typographical errors.

Reviewer: Andras Batkai (Roma)

##### MSC:

47D06 | One-parameter semigroups and linear evolution equations |

47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |

93B35 | Sensitivity (robustness) |

35B40 | Asymptotic behavior of solutions to PDEs |

46A30 | Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness) |