On two theorems regarding exponential stability
The author considers the exponential stability of linear skew-evolution semiflows, generated by a class of evolution equations based on the results of \textit{E. A. Barbashin} [Introduction to the theory of stability (Russian). Moskva: Nauka (1967; Zbl 0155.13501)] and \textit{R. Datko} [J. Math. Anal. Appl. 32, 610--616 (1970; Zbl 0211.16802)] in the theory of stability of dynamical systems.NEWLINENEWLINE The goal is to present a unified approach for both discrete-time versions of Barbashin's condition and uniform Datko's condition extending the results given by the author [Appl. Anal. 90, No. 11--12, 1897--1907 (2011; Zbl 1233.34025)], \textit{K. M. Przyłuski} and \textit{S. Rolewicz} [Syst. Control Lett. 4, 307--315 (1984; Zbl 0543.93057)] and \textit{J. Zabczyk} [SIAM J. Control 12, 721--735 (1974; Zbl 0254.93027)].NEWLINENEWLINE As an illustration, he gives the formulation of one of the obtained statements: The linear skew-evolution semiflow \((\Phi,\sigma)\) is uniformly exponentially stable if and only if there exist \(K>0\) and a continuous, nondecreasing function \(b:\mathbb R^+\to\mathbb R^+\), \(b(0)=0\), \(b(t)>0\), \(t>0\), such that NEWLINE\[NEWLINE \sup\int_0^nb(| \Phi(n+\tau+m,\tau+m,\sigma(\tau+m,m,\theta))| )\,d\tau<K<\infty. NEWLINE\]
scientific article; zbMATH DE number 6217156