On the exponential stability of a dynamical system on a time scale
The article deals with dynamical systems on a time scale. A time scale \({\mathbb T}\) is an arbitrary nonempty closed subset of the set \({\mathbb R}\). A function \(F:\;{\mathbb T} \to {\mathbb R}\) is called \(\Delta\)-differentiable at \(t \in {\mathbb T}^\kappa\) if, for any \(\varepsilon > 0\), there exists \(\gamma \in {\mathbb R}\) and a \(W\)-neighborhood of \(t \in {\mathbb T}^\kappa\) such that \(|[f(\sigma(t)) - f(s)] - \gamma[\sigma(t) - s]| < \varepsilon|\sigma(t) - s|\) for all \(s \in W\); in this case, \(f^\Delta(t) = \gamma\). Here, \(\sigma(t) = \inf \{s \in {\mathbb T}:\;s > t\}\) (and analogously, \(\rho(t) = \sup \{s \in {\mathbb T}:\;s < t\}\)), \[ {\mathbb T}^\kappa := \begin{cases}{\mathbb T} \setminus \rho(\sup {\mathbb T}),\sup {\mathbb T}) & \text{if} \;\;\sup {\mathbb T} < \infty, \\ {\mathbb T} & \text{if} \;\;\sup {\mathbb T} = \infty.\end{cases} \] An analogue of the Newton-Leibniz formula is proved and a theorem about \(\Delta\)-differentiability for the composition of two functions. Further, an analog of the exponential function is defined, and an analog of the formula for solving linear equation \[ x^\Delta(t) = p(t)x(t) + f(t) \] is obtained. The main part of the article is devoted to equations of the type \[ x^\Delta(t) = f(t,x(t)),\tag{1} \] where \(x(t): {\mathbb T} \to {\mathbb R}^n\). The main result here is a theorem about conditions under which the equilibrium of system (1) is exponentially stable or uniformly exponential stable. At the end of the article, an example of a two-dimensional 'quasi-linear' system is considered.
scientific article; zbMATH DE number 5905417