Finite-time stability of fractional delayed neural networks

In this paper, finite-time stability of a class of fractional delayed neural networks of retarded-type with commensurate order between 0 and 1 is investigated. For such problems in integer-order systems, Lyapunov functions are usually constructed, whereas no specific Lyapunov functions exist in fractional-order cases. By employing inequalities such as Holder inequality, Gronwall inequalities and inequality scaling skills, some finite-time stability results are derived. For fractional delayed neural models of retarded-type with order 0 < α < 0.5 and 0.5 ? α < 1 , sufficient conditions for the finite-time stability are presented. Numerical simulations also verify the theoretical results.

[1]  Mohammad Bagher Menhaj,et al.  Fractional-Order Hopfield Neural Networks , 2008, ICONIP.

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

[3]  P. Dorato.  SHORT-TIME STABILITY IN LINEAR TIME-VARYING SYSTEMS , 1961 .

[4]  Tingwen Huang,et al.  A mode-dependent approach to state estimation of recurrent neural networks with Markovian jumping parameters and mixed delays , 2013, Neural Networks.

[5]  Hiroyuki Ichihara,et al.  Finite-time control for linear systems with impulse control , 2009, 2009 ICCAS-SICE.

[6]  Christophe Farges,et al.  On stability of commensurate fractional Order Systems , 2012, Int. J. Bifurc. Chaos.

[7]  Ivo Petrás,et al.  A Note on the Fractional-Order Cellular Neural Networks , 2006, The 2006 IEEE International Joint Conference on Neural Network Proceedings.

[8]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[9]  Gianmaria De Tommasi,et al.  Sufficient Conditions for Finite-Time Stability of Impulsive Dynamical Systems , 2009, IEEE Transactions on Automatic Control.

[10]  P. Arena,et al.  Bifurcation and Chaos in Noninteger Order Cellular Neural Networks , 1998 .

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

[12]  P. Butzer,et al.  AN INTRODUCTION TO FRACTIONAL CALCULUS , 2000 .

[13]  Francesco Amato,et al.  Finite-time stability of linear time-varying systems with jumps , 2009, Autom..

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

[15]  Carlo Cosentino,et al.  Finite-time stabilization via dynamic output feedback, , 2006, Autom..

[16]  A. Fairhall,et al.  Fractional differentiation by neocortical pyramidal neurons , 2008, Nature Neuroscience.

[17]  Attila Gilányi,et al.  An Introduction to the Theory of Functional Equations and Inequalities , 2008 .

[18]  SACHIN BHALEKAR,et al.  A PREDICTOR-CORRECTOR SCHEME FOR SOLVING NONLINEAR DELAY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 2011 .

[19]  P. Arena,et al.  Chaotic behavior in noninteger-order cellular neural networks , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Lihong Huang,et al.  Finite time stability of periodic solution for Hopfield neural networks with discontinuous activations , 2013, Neurocomputing.

[21]  Yong Zhou,et al.  New concepts and results in stability of fractional differential equations , 2012 .

[22]  Sophie Tarbouriech,et al.  Finite-Time Stabilization of Linear Time-Varying Continuous Systems , 2009, IEEE Transactions on Automatic Control.

[23]  Alain Oustaloup,et al.  Stability of Closed Loop fractional Order Systems and Definition of damping Contours for the Design of Controllers , 2012, Int. J. Bifurc. Chaos.

[24]  Tiedong Ma,et al.  Dynamic analysis of a class of fractional-order neural networks with delay , 2013, Neurocomputing.

[25]  Yong Xu,et al.  Chaos in a Fractional-Order Dynamical Model of Love and Its Control , 2011, NL-MUA.

[26]  Haijun Jiang,et al.  Α-stability and Α-synchronization for Fractional-order Neural Networks , 2012, Neural Networks.

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

[28]  P. Dorato,et al.  Finite time stability under perturbing forces and on product spaces , 1967, IEEE Transactions on Automatic Control.

[29]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[30]  N. I. Morozov,et al.  Stability of motion over a finite interval of time , 1978 .

[31]  Matthias Schroder.  Principles Of Differential And Integral Equations , 2016 .

[32]  J. R. Raol.  Parameter estimation of state space models by recurrent neural networks , 1995 .

[33]  Jianying Yang,et al.  Dynamical models of happiness with fractional order , 2010 .

[34]  D. J. Marsh,et al.  Robust Nonlinear Autoregressive Moving Average Model Parameter Estimation Using Stochastic Recurrent Artificial Neural Networks , 2004, Annals of Biomedical Engineering.

[35]  吴然超,et al.  Finite-Time Stability of Fractional-Order Neural Networks with Delay＊ , 2013 .

[36]  Randall D. Beer,et al.  Parameter Space Structure of Continuous-Time Recurrent Neural Networks , 2006, Neural Computation.

[37]  Zhen Wang,et al.  Chaos and hyperchaos in fractional-order cellular neural networks , 2012, Neurocomputing.

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

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

[40]  Mohammad Saleh Tavazoei,et al.  A note on the stability of fractional order systems , 2009, Math. Comput. Simul..

[41]  Francesco Amato,et al.  Finite-Time Stability of Linear Time-Varying Systems: Analysis and Controller Design , 2010, IEEE Transactions on Automatic Control.