Abstract
Fractional differential equations with impulses arise in modeling real world phenomena where the state changes instantaneously at some moments. Often, these instantaneous changes occur at random moments. In this situation the theory of Differential equations has to be combined with Probability theory to set up the problem correctly and to study the properties of the solutions. We study the case when the time between two consecutive moments of impulses is exponentially distributed. In connection with the application of the Riemann–Liouville fractional derivative in the equation, we define in an appropriate way both the initial condition and the impulsive conditions. We consider the case when the lower limit of the Riemann–Liouville fractional derivative is fixed at the initial time. We define the so called p-moment Mittag–Leffler stability in time of the model. In the case of integer order derivative the introduced type of stability reduces to the p–moment exponential stability. Sufficient conditions for p–moment Mittag–Leffler stability in time are obtained. The argument is based on Lyapunov functions with the help of the defined fractional Dini derivative. The main contributions of the suggested model is connected with the implementation of impulses occurring at random times and the application of the Riemann–Liouville fractional derivative of order between 0 and 1. For this model the p-moment Mittag–Leffler stability in time of the model is defined and studied by Lyapunov functions once one defines in an appropriate way their Dini fractional derivative.
Highlights
Fractional differential equations are considered as a generalization of ordinary differential equations and many results about different types of fractional differential equations are obtained in the literature [1,2,3]. this is not the situation with fractional impulsive differential equations because of the nonlocal feature
In connection with the presence of the RL derivative, we define in an appropriate way both the impulsive conditions and the initial condition
The case of impulses occurring at random times is studied when the waiting time between two consecutive impulses is exponentially distributed; The statement of the initial value problem with Riemann–Liouville fractional derivatives of order between 0 and 1 is given in an appropriate way; The p-moment Mittag–Leffler stability in time of the model is defined; The fractional Dini derivative of the Lyapunov function is defined; Sufficient conditions for p-moment Mittag–Leffler stability in time are obtained
Summary
Fractional differential equations are considered as a generalization of ordinary differential equations and many results about different types of fractional differential equations are obtained in the literature [1,2,3]. The presence of the RL fractional derivative and its singularity at the initial time leads to excluding this point from the interval of the stability and we define a new type of stability called the p-moment Mittag–Leffler stability in time We study this type of stability by employing Lyapunov functions. The case of impulses occurring at random times is studied when the waiting time between two consecutive impulses is exponentially distributed; The statement of the initial value problem with Riemann–Liouville fractional derivatives of order between 0 and 1 is given in an appropriate way; The p-moment Mittag–Leffler stability in time of the model is defined; The fractional Dini derivative of the Lyapunov function is defined; Sufficient conditions for p-moment Mittag–Leffler stability in time are obtained
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