Stability of Caputo fractional differential equations with non-instantaneous impulses (Q2818499)

The authors consider the initial value problem for the following system of fractional differential equations with Caputo-type derivative: NEWLINE\[NEWLINE\phantom{}_{\tau_k}^{c}D^qx=f(t,x)\text{, }t\in\left(\tau_k,t_{k+1}\right]\text{, }k\in\mathbb{N}_0NEWLINE\]NEWLINE subject to \(x(t)=\phi_i\big(t,x(t_i-0)\big)\) for \(t\in\left(t_i,s_i\right]\), \(k\in\mathbb{N}\), and \(x\big(t_0\big)=x_0\), where \(\{t_i\}\) and \(\{s_i\}\) are two increasing sequences with \(s_0=0<t_i\leq s_i<t_{i+1}\). This is an example of a fractional differential equation with noninstantaneous impulses. The authors consider only the case in which \(q\in(0,1)\). The stability of this system is studied using Lyapunov-like functions. Several examples are provided and a preliminary discussion of impulsive differential equations is also included for the convenience of the reader.