Degenerate semi-group methods for the exponential stability of the first order singular distributed parameter systems
This paper deals with the exponential stability of degenerate semigroups in Hilbert spaces. A degenerate semigroup \((T(t))_{t\geq 0}\) is a family of bounded linear operators on a Banach space \(X\) such that \(T(t+s)=T(t)T(s),\,t,s\geq 0\), \(T(\cdot)\) is strongly continuous in \(\mathbb{R}_+\) and \(T(0)=P\) a bounded linear operator having nonempty kernel. This paper contains two results. The first one is: ``Exponential stability of a degenerate semigroup \(T(\cdot)\) is equivalent to \(\omega_0:=\inf_{t>0}\frac{\ln \|T(t)\|}{t}<0\).'' This is well-known for \(C_0\)-semigroups and the proof remains true (without any changes) for degenerate semigroups cf. Proposition IV.2.2 in the book of [\textit{K. J. Engel} and \textit{R. Nagel} [One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics. 194. Berlin: Springer (2000; Zbl 0952.47036)]). The second result is Datko's theorem for degenerate semigroups. Also, in this case, the proof of the classical Datko's theorem remains exactly the same for degenerate semigroups (cf. Theorem V.1.8 in [loc. cit.)]) and it is shorter and more elegant than the one given by the authors of this paper.
first order singular distributed parameter systems
scientific article; zbMATH DE number 5826122
scientific article