Stability analysis of nonlinear Hadamard fractional differential system

Abstract The stability of the zero solution of a class of nonlinear Hadamard type fractional differential system is investigated by utilizing a new fractional comparison principle. The novelty of this paper is based on some new fractional differential inequalities along the given nonlinear Hadamard fractional differential system. A comparison principle employing the new fractional differential inequality for scalar Hadamard fractional differential system is presented. Based on the new comparison principle, some sufficient conditions for the (generalized) stability and the (generalized) Mittag-Leffler stability are given.

[1]  S. Leela,et al.  LYAPUNOV THEORY FOR FRACTIONAL DIFFERENTIAL EQUATIONS , 2008 .

[2]  Ahmed Alsaedi,et al.  A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions , 2016 .

[3]  Ravi P. Agarwal,et al.  Practical Stability of Caputo fractional differential equations by Lyapunov functions , 2016 .

[4]  Donal O’Regan,et al.  A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations , 2016 .

[5]  Dumitru Baleanu,et al.  Explicit iteration to Hadamard fractional integro-differential equations on infinite domain , 2016 .

[6]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[7]  Igor Podlubny,et al.  Mittag-Leffler stability of fractional order nonlinear dynamic systems , 2009, Autom..

[8]  Mathieu Moze,et al.  LMI stability conditions for fractional order systems , 2010, Comput. Math. Appl..

[9]  K. Moore,et al.  Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[10]  Anatoly A. Kilbas,et al.  HADAMARD-TYPE FRACTIONAL CALCULUS , 2001 .

[11]  V. Kiryakova.  Generalized Fractional Calculus and Applications , 1993 .

[12]  Aleksandar M. Spasic,et al.  Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach , 2009, Math. Comput. Model..

[13]  J. Vasundhara Devi,et al.  Variational Lyapunov method for fractional differential equations , 2012, Comput. Math. Appl..

[14]  Guotao Wang,et al.  On a Hadamard-type fractional turbulent flow model with deviating arguments in a porous medium , 2017 .

[15]  Donal O’Regan,et al.  Stability of Caputo fractional differential equations by Lyapunov functions , 2015 .

[16]  V. Lakshmikantham,et al.  Theory of Fractional Dynamic Systems , 2009 .

[17]  Margarita Rivero,et al.  Stability of Fractional Order Systems , 2013 .

[18]  Yangquan Chen,et al.  Computers and Mathematics with Applications Stability of Fractional-order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–leffler Stability , 2022 .

[19]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[20]  Juan J. Trujillo,et al.  Hadamard-type integrals as G-transforms , 2003 .

[21]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

[22]  Guotao Wang,et al.  Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain , 2017, Appl. Math. Comput..

[23]  Jiang Wei.  Finite-time Stability of Linear Fractional Order Neutral Systems , 2011 .

[24]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[25]  Bashir Ahmad,et al.  A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations , 2014 .

[26]  Danna Zhou,et al.  d. , 1934, Microbial pathogenesis.

[27]  Xian-Feng Zhou,et al.  Mittag–Leffler stability of nonlinear fractional neutral singular systems , 2012 .

[28]  Yong Zhou,et al.  Existence and stability of fractional differential equations with Hadamard derivative , 2013 .

[29]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[30]  A. Ranjbar,et al.  SOME FRACTIONAL COMPARISON RESULTS AND STABILITY THEOREM FOR FRACTIONAL TIME DELAY SYSTEMS , 2013 .

[31]  I. Podlubny.  Fractional differential equations , 1998 .

[32]  Junwei Wang,et al.  Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative , 2015, Appl. Math. Comput..

[33]  Changpin Li,et al.  A survey on the stability of fractional differential equations , 2011 .

[34]  Federico Polito,et al.  On some operators involving Hadamard derivatives , 2013, 1304.1058.

[35]  Wei Jiang,et al.  Stability of nonlinear Caputo fractional differential equations , 2016 .

[36]  Ravi P. Agarwal,et al.  Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line , 2018, J. Comput. Appl. Math..

[37]  Paul L. Butzer,et al.  Compositions of Hadamard-type fractional integration operators and the semigroup property , 2002 .