Stability analysis of nonlinear fractional-order systems with variable-time impulses

This paper aims at analyzing the stability analysis for a class of variable-time impulsive fractional-order nonlinear systems. Based on the theory of fractional calculus, the theory of impulsive differential equation, inequality techniques, and the B-equivalence method, the variable-time jump operator of the considered system can be updated as a fixed-time substitution, and the fractional-order system with the latter operator can be regarded as the comparison system of the original system. In addition, both graphic illustration and theoretical explanation are presented. Finally, two numerical examples are shown to demonstrate the validity and feasibility of the obtained results.

[1]  Michal Fečkan,et al.  Stability analysis of impulsive fractional-order systems by vector comparison principle , 2015 .

[2]  Eva Kaslik,et al.  Nonlinear dynamics and chaos in fractional-order neural networks , 2012, Neural Networks.

[3]  Ivanka M. Stamova,et al.  Integral manifolds of impulsive fractional functional differential systems , 2014, Appl. Math. Lett..

[4]  Weihua Deng,et al.  Remarks on fractional derivatives , 2007, Appl. Math. Comput..

[5]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[6]  Jun-Guo Lu,et al.  Robust Stability and Stabilization of Fractional-Order Interval Systems with the Fractional Order $\alpha$: The $0≪\alpha≪1$ Case , 2010, IEEE Transactions on Automatic Control.

[7]  Gang Feng,et al.  Consensus Analysis Based on Impulsive Systems in Multiagent Networks , 2012, IEEE Transactions on Circuits and Systems I: Regular Papers.

[8]  Manuel A. Duarte-Mermoud,et al.  Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems , 2015, Commun. Nonlinear Sci. Numer. Simul..

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

[10]  Cheng Hu,et al.  Global stability problem for feedback control systems of impulsive fractional differential equations on networks , 2015, Neurocomputing.

[11]  S. Dashkovskiy,et al.  Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods , 2010, 1011.2865.

[12]  Marat Akhmet,et al.  Principles of Discontinuous Dynamical Systems , 2010 .

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

[14]  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 .

[15]  Qiankun Song,et al.  Stabilization and synchronization of chaotic systems with mixed time-varying delays via intermittent control with non-fixed both control period and control width , 2015, Neurocomputing.

[16]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

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

[18]  Manuel A. Duarte-Mermoud,et al.  Lyapunov functions for fractional order systems , 2014, Commun. Nonlinear Sci. Numer. Simul..

[19]  Yong Zhou,et al.  On the concept and existence of solution for impulsive fractional differential equations , 2012 .

[20]  V. Lakshmikantham,et al.  Comparison principle for impulsive differential equations with variable times and stability theory , 1994 .

[21]  Jinde Cao,et al.  Pinning impulsive stabilization of nonlinear Dynamical Networks with Time-Varying Delay , 2012, Int. J. Bifurc. Chaos.

[22]  JinRong Wang,et al.  On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives , 2015, Appl. Math. Lett..

[23]  Enes Yilmaz.  Almost periodic solutions of impulsive neural networks at non-prescribed moments of time , 2014, Neurocomputing.

[24]  Carlos Silvestre,et al.  Stability of networked control systems with asynchronous renewal links: An impulsive systems approach , 2013, Autom..

[25]  Tao Yang,et al.  In: Impulsive control theory , 2001 .

[26]  Johnny Henderson,et al.  Impulsive functional differential equations with variable times , 2004 .

[27]  Ivanka M. Stamova,et al.  Stability Analysis of Impulsive Functional Differential Equations , 2009 .

[28]  Mustafa Sayli,et al.  Global robust asymptotic stability of variable-time impulsive BAM neural networks , 2014, Neural Networks.

[29]  Wei Jiang,et al.  Impulsive fractional functional differential equations , 2012, Comput. Math. Appl..

[30]  V. Lakshmikantham,et al.  Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times , 1994 .

[31]  Xinghuo Yu,et al.  Asynchronous impulsive containment control in switched multi-agent systems , 2016, Inf. Sci..

[32]  Jun-Guo Lu,et al.  Stability Analysis of a Class of Nonlinear Fractional-Order Systems , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[33]  Marat Akhmet.  On the General Problem of Stability for Impulsive Differential Equations , 2003 .

[34]  Ivanka M. Stamova,et al.  Stability analysis of impulsive functional systems of fractional order , 2014, Commun. Nonlinear Sci. Numer. Simul..

[35]  D. Willett.  Nonlinear vector integral equations as contraction mappings , 1964 .