ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

P-Moment Exponential Stability of Caputo Fractional Differential Equations with Random Impulses

Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 49--63 | DOI:10.5890/DNC.2017.03.005

Ravi Agarwal$^{1}$, Snezhana Hristova$^{2}$, Donal O’Regan$^{3}$

$^{1}$ Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA

$^{2}$ Plovdiv University, Tzar Asen 24, Plovdiv 4000, Bulgaria

$^{3}$ School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Abstract

Fractional differential equations with random impulses arise in modeling real world phenomena where the state changes instantaneously at uncertain moments. Using queuing theory and the usual distribution for waiting time, we study the case of exponentially distributed random variables between two consecutive moments of impulses. The p-moment exponential stability of solutions is defined and studied when the waiting time between two consecutive impulses is exponentially distributed. The argument is based on Lyapunov functions. We discuss both continuous and differentiable Lyapunov functions and Caputo fractional Dini derivatives and Caputo derivatives are applied. Some examples are given to illustrate our results.

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