Pötzsche, Christian; Siegmund, Stefan; Wirth, Fabian A spectral characterization of exponential stability for linear time-invariant systems on time scales. (English) Zbl 1054.34086 Discrete Contin. Dyn. Syst. 9, No. 5, 1223-1241 (2003). The authors define various concepts of exponential stability (exponential, uniform exponential, robust exponential stability) for the linear dynamic system \(x^\Delta=A(t)\,x\) on an arbitrary time scale \({\mathbb T}\) (nonempty closed subset of \({\mathbb R}\)). In the autonomous case for differential (\({\mathbb T}={\mathbb R}\)) and difference (\({\mathbb T}={\mathbb Z}\)) systems these concepts coincide. However, provided examples illustrate that in the general time scale setting these notions do not need to coincide even for time-invariant systems. For the scalar equation \(x^\Delta=\lambda x\), the authors derive a characterization of its exponential stability, and define the set of exponential stability. This set is then explicitly calculated for several examples. Higher-dimensional systems are treated for the case of Jordan reducible matrices \(A(t)\). In this setting and for the autonomous case \(x^\Delta=Ax\), a characterization of the exponential stability is proven under the assumption of a uniform regressivity of the eigenvalues of \(A\). In the nonregressive case, the latter assumption is replaced by a uniform exponential stability of a scalar equation associated with the defective eigenvalues (i.e., the eigenvalues having different geometric and algebraic multiplicities).This nice paper will be useful for researchers interested in stability criteria for linear differential/difference/dynamic equations/systems. Reviewer: Roman Hilscher (Brno) Cited in 4 ReviewsCited in 68 Documents MSC: 34D20 Stability of solutions to ordinary differential equations 39A10 Additive difference equations 34A30 Linear ordinary differential equations and systems 39A12 Discrete version of topics in analysis Keywords:time scale; linear dynamic equation/system; exponential stability; regressive matrix PDFBibTeX XMLCite \textit{C. Pötzsche} et al., Discrete Contin. Dyn. Syst. 9, No. 5, 1223--1241 (2003; Zbl 1054.34086) Full Text: DOI