A spectral characterization of exponential stability for linear time-invariant systems on time scales
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.
scientific article; zbMATH DE number 1925730