Abstract
One approach to study various stability properties of solutions of nonlinear Caputo fractional differential equations is based on using Lyapunov like functions. A basic question which arises is the definition of the derivative of the Lyapunov like function along the given fractional equation. In this paper, several definitions known in the literature for the derivative of Lyapunov functions among Caputo fractional differential equations are given. Applications and properties are discussed. Several sufficient conditions for stability, uniform stability and asymptotic stability with respect to part of the variables are established. Several examples are given to illustrate the theory.
Highlights
Fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering since the behavior of many systems, such as physical phenomena having memory and genetic characteristics, can be adequately modeled by fractional differential systems
The question of stability is of interest in physical and biological systems, such as the fractional Duffing oscillator [4], fractional predator-prey and rabies models [5], etc. and stability theory of fractional differential equations(FDEs) is widely applied to chaos and chaos synchronization [6] because of its potential applications in control processing and secure communication
Several results were obtained such as structural stability of systems with Riemann–Liouville derivative [10]), continuous dependence of solution on initial conditions for Caputo nonautonomous fractional differential equations [11], stability in the sense of Lyapunov by using Gronwall’s inequality and Schwartz’s inequality [12], Mittag–Leffler stability [13], and local asymptotic stability [14]
Summary
Fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering since the behavior of many systems, such as physical phenomena having memory and genetic characteristics, can be adequately modeled by fractional differential systems (see, for example, [1,2,3]). A Caputo fractional Dini derivative of a Lyapunov function among nonlinear Caputo fractional differential equations is presented. This type of derivative was introduced in [16] and used to study stability and asymptotic stability [16] of Caputo fractional differential equations, and for stability of Caputo fractional differential equations with non-instantaneous impulses [17]. Comparison results using this definition and scalar fractional differential equations are presented and several sufficient conditions for stability, uniform stability, asymptotic stability with respect to part of the variables are given.
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