Stability of abstract dynamic equations on time scales by Lyapunov's second method

Authors

ALAA HAMZA
KARIMA ORABY

Abstract

In this paper, we use the Lyapunov's second method to obtain new sufficient conditions for many types of stability like exponential stability, uniform exponential stability, $h$-stability, and uniform $h$-stability of the nonlinear dynamic equation \begin{equation*} x^{\Delta}(t)=A(t)x(t)+f(t,x),\;t\in \mathbb{T}^+_\tau:=[\tau,\infty)_{\mathbb T}, \end{equation*} on a time scale $\mathbb T$, where $A\in C_{rd}(\mathbb T,L(X))$ and $f:\mathbb T\times X\to X$ is rd-continuous in the first argument with $f(t,0)=0.$ Here $X$ is a Banach space. We also establish sufficient conditions for the nonhomogeneous particular dynamic equation \begin{equation*} x^{\Delta}(t)=A(t)x(t)+f(t),\,t\in\mathbb{T}^+_{\tau}, \end{equation*} to be uniformly exponentially stable or uniformly $h$-stable, where $f\in C_{rd}(\mathbb T,X)$, the space of rd-continuous functions from $\mathbb T$ to $X$. We construct a Lyapunov function and we make use of this function to obtain our stability results. Finally, we give illustrative examples to show the applicability of the theoretical results.

DOI

10.3906/mat-1703-65

Keywords

Lyapunov stability theory, dynamic equations, time scales

First Page

841

Last Page

861

Recommended Citation

HAMZA, ALAA and ORABY, KARIMA (2018) "Stability of abstract dynamic equations on time scales by Lyapunov's second method," Turkish Journal of Mathematics: Vol. 42: No. 3, Article 9. https://doi.org/10.3906/mat-1703-65
Available at: https://journals.tubitak.gov.tr/math/vol42/iss3/9