Caputo Fractional Dierential Equations with Non-Instantaneous Impulses and Strict Stability by Lyapunov Functions

Abstract

In this paper the statement of initial value problems
for fractional differential equations with non-instantaneous
impulses is given. These equations are adequate models for phenomena
that are characterized by impulsive actions starting at arbitrary
fixed points and remaining active on finite time intervals. Strict
stability properties of fractional differential equations with
non-instantaneous impulses by the Lyapunov approach is studied. An
appropriate definition (based on the Caputo fractional Dini
derivative of a function) for the derivative of Lyapunov functions
among the Caputo
fractional differential equations with non-instantaneous impulses is presented.
Comparison results using this definition and scalar fractional
differential equations with non-instantaneous impulses are presented
and sufficient conditions for strict stability and uniform strict
stability are given. Examples are given to illustrate the theory.