Discontinuity, Nonlinearity, and Complexity
PMoment Exponential Stability of Caputo Fractional Differential Equations with Random Impulses
Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 4963  DOI:10.5890/DNC.2017.03.005
Ravi Agarwal$^{1}$, Snezhana Hristova$^{2}$, Donal O’Regan$^{3}$
$^{1}$ Department of Mathematics, Texas A&M UniversityKingsville, 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
Download Full Text PDF
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 pmoment 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.
References

[1]  Agarwal R. and Hristova, S. (2012), Strict stability in terms of two measures for impulsive differential equations with supremum, Appl. Anal., 91, 13791392. 

[2]  Hristova S. (2010), Integral stability in terms of two measures for impulsive functional differential equations, Math. Comput. Modell., 51, 100108. 

[3]  Hristova S. (2010), Stability on a cone in terms of two measures for impulsive differential equations with supremum , Appl. Math. Lett., 23, 5, 508511. 

[4]  Hristova S. (2009), Razumikhin method and cone valued Lyapunov functions for impulsive differential equations with supremum, Intern. J. Dynam. Syst. Diff. Eq., 2, 34, 223236. 

[5]  Hristova S., Stefanova K. (2012), Practical stability of impulsive differential equations with supremum by integral inequalities, Eur. J. Pure Appl. Math., 5, 1, 3044. 

[6]  Hristova S. (2010), Lipschitz stability for impulsive differential equations with supremum, Intern. Electr. J. Pure Appl. Math., 1, 4, 345358. 

[7]  Hristova S. (2009), Qualitative Investigations and Approximate Methods for Impulsive Differential Equations, Nova Sci. Publ. 

[8]  LakshmikanthamV., Bainov D.D. , Simeonov P.S. (1989), Theory of Impulsive Differential Equations,World Scientific, Singapore. 

[9]  SanzSerna J.M., Stuart A.M. (1999), Ergodicity of dissipative differential equations subject to random impulses, J. Diff. Equ., 155, 262284. 

[10]  Wu S., Hang D , Meng X. (2004), pMoment Stability of Stochastic Equations with Jumps, Appl. Math. Comput., 152, 505519. 

[11]  Bagley, R.L. and Calico, R.A. (1991), Fractional order state equations for the control of viscoelasticallydamped structures, J. Guid., Contr. Dyn., 14(2), 304311. 

[12]  Laskin N. (2000), Fractional market dynamics, Phys. A, Stat. Mech. Appl., 287, 34, 482492. 

[13]  Anguraj, A., and Vinodkumar, A. (2010), Existence, uniqueness and stability results of random impulsive semilinear differential systems, Nonlinear Anal. Hybrid Syst., 3, 475483. 

[14]  Anguraj, A., Ranjini, M.C., Rivero, M., and Trujillo, J. J. (2015), Existence results for fractional neutral functional differential equations with random impulses, Mathematics, 2015(3), 1628. 

[15]  Wang J.R., Feckan M., Zhou Y. (2016), Random Noninstantaneous Impulsive Models for Studying Periodic Evolution Processes in Pharmacotherapy, Mathematical Modeling and Applications in Nonlinear Dynamics, 14, Nonlinear Systems and Complexity, 87107. 

[16]  Lakshmikantham V., Leela S., Devi J.V. (2009), Theory of Fractional Dynamical Systems, Cambridge Scientific Publishers. 

[17]  Podlubny I. (1999), Fractional Differential Equations, Academic Press, San Diego. 

[18]  Das Sh. ( 2011), Functional Fractional Calculus, SpringerVerlag Berlin Heidelberg. 

[19]  Diethelm K. (2010), The Analysis of Fractional Differential Equations, SpringerVerlag Berlin Heidelberg. 

[20]  Devi J. V., Mc Rae F.A., Drici Z. (2010), Variational Lyapunov method for fractional differential equations, Comput. Math. Appl. 64, 29822989. 

[21]  AguilaCamacho, N., DuarteMermoud, M. A., and Gallegos, J. A. (2014), Lyapunov functions for fractional order systems, Comm. Nonlinear Sci. Numer. Simul., 19, 29512957. 

[22]  Baleanu D., Mustafa O.G. (2010), On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl. 59, 18351841. 

[23]  Agarwal, R., Benchohra, M., and Slimani, B. A. (2008), Existence results for differential equations with fractional order and impulses, Mem. Differ. Equ. Math. Phys, 44, 121. 

[24]  Ahmad, B. and Sivasundaram, S. (2009), Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid Syst. 3, 251258. 

[25]  Benchohra M., Slimani B. A. (2009), Existence and uniqueness of solutions to impulsive fractional differential equations, Electronic Journal of Differential Equations, No. 10, 111. 

[26]  Wang G., Ahmad B. , Zhang L., Nieto J. (2014), Comments on the concept of existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat. 19, 401403. 

[27]  Feckan M., Zhou Y., Wang J. (2012), On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17, 30503060. 

[28]  Devi J. V., Mc Rae F.A., Drici Z. (2010), Generalized quasilinearization for fractional differential equations, Comput. Math. Appl. 59, 10571062. 

[29]  DuarteMermoud M. A., AguilaCamacho N., Gallegos J. A., CastroLinares R. (2015), Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Comm. Nonlinear Sci. Numer. Simul., 22, 650659. 

[30]  Agarwal, R., Hristova, S., and O’Regan, D. (2015), Lyapunov functions and strict stability of Caputo fractional differential equations, Adv. Diff. Eq., 2015. 

[31]  Agarwal R., O’Regan D., Hristova S. (2015), Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math., 60, 6, 653676. 

[32]  Agarwal, R., Hristova S., and O’Regan, D. (2016), Practical stability of Caputo fractional differential equations by Lyapunov functions, Diff. Eq. Appl. 8, 1, 5368. 