1到2分数阶非线性动力系统的稳定性分析（英文）
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摘要
One new theorem for Caputo fractional derivative and two new theorems for Caputo fractional order systems, when 1 < a < 2, are proposed in this paper. The results have proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the instability and the stability of many fractional order systems,which can be nonlinear and time varying.
One new theorem for Caputo fractional derivative and two new theorems for Caputo fractional order systems, when 1 < a < 2, are proposed in this paper. The results have proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the instability and the stability of many fractional order systems,which can be nonlinear and time varying.
One new theorem for Caputo fractional derivative and two new theorems for Caputo fractional order systems, when 1 < a < 2, are proposed in this paper. The results have proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the instability and the stability of many fractional order systems,which can be nonlinear and time varying.
引文
[1]HAN R J,GE J S,JIANG W.Existence of Positive Solutions of Three-point Boundary Value Problem for Higher Order Nonlinear Fractional Differential Equations[J].Chin,Quart,J,of Math,2012,27(4):516-525.
[2]SI J F,JIANG W.Sliding Mode Control for Fractional Differential Systems with State-delay[J].Chin,Quart,J,of Math,2012,27(1):117-122.
[3]HUANG Z M,XIE Z S,HOU C M.On a Discrete Fractional Boundary Value Problem with Nonlocal Fractional Boundary Conditions[J].Chin,Quart,J,of Math,2014,29(4):539-552.
[4]LI Y,JIANG W,HU B B.Stability of Neutral Fractional Neural Networks with Delay[J].Chin,Quart,J,of Math,2016,31(4):422-429.
[5]LI X Y,XIANG J R,WU Y Y.Laplace Transform Method Applied to Solve Fractional Fractional Equations[J].Chin Quart J of Math,2015,30(1):121-129.
[6]MATIGNON D.Stability results for fractional differential equations with applications to control processing,in:IMACS-SMCProceedings,Lille,France,1996.
[7]GABANO J D,POINOT T.Fractional modelling and identification of thermal systems[J].Signal Process,2011,91:531-541.
[8]LIN J,POINOT T,TRIGEASSOU JC,KABBAJ H,FAUCHER J.Modelisation et identification d,ordre non entire d,une machine asynchrone.In:Conference Internationale Francophone d,Automatique 2000.
[9]LI Y,CHEN Y,PODLUBNY I.Stability of fractional-order nonlinear dynamic systems:Lyapunov direct method and generalized Mittag Leffer stability[J].Comput Math Appl,2010,59:1810-1821.
[10]MANUEL A,DUARTE M,NORELYS A C.Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems[J].Commun Nonlinear Sci Numer Simulat,2015,22:650-659.
[11]NORELYS A C,MANUEL A,DUARTE M.Lyapunov functions for fractional order systems.Commun Nonlinear Sci Numer Simulat,2014,19:2951-2957.
[12]ANAYA J F,ANTONIO J N,GALANTE J J.Lyapunov functions for a class of nonlinear systems using Caputo derivative[J].Commun Nonlinear Sci Numer Simulat,2017,43:91-99.
[13]JAVIER A,GALLEGOS M A,DUARTE M.On the Lyapunov Theory for fractional order systems[J].Applied Mathematics and Computation,2016,287:161-170.
[14]LI C P,ZHANG F R.A survey on the stability of fractional differential equations[J].European Physical Journal Special Topics,2011,193:27-47.
[15]SONG L,WU R C,CHEN L P.Comparison principles and stability of nonlinear fractional-order cellular neural networks with multiple time delays.Neurocomputing,2015,168:618-625.
[16]ZHANG L,SONG Q K,ZHAO Z J.Stability analysis of fractional-order complex-valued neural networks with both leakage and discrete delays.Applied Mathematics and Computation,2017,298:296-309.
[17]HU J B,ZHAO L D,LU G P,ZHANG S B.The stability and control of fractional nonlinear system with distributed time delay,Applied Mathematical Modelling,2016,40:3257-3263.
[18]SABATIER J,MOZE M,FARGES C.LMI stability conditions for fractional order systems[J].Computers and Mathematics with Applications,2010,59:1594-1609.
[19]GUO B L.Fractional Partial Differential Equations and their Numerical Solution[M].Science Press of China,2011.
[20]KILBAS A,SRIVASTAVE H,TRUJILLO J.Theory and Applications of Fractional Differential Equations[M].Elsevier,North Holland 2006.
[21]PODLUBNY I.Fractional Differential Equations[M].Technical University of Kosice,Slovak Republic,1999.
[2]SI J F,JIANG W.Sliding Mode Control for Fractional Differential Systems with State-delay[J].Chin,Quart,J,of Math,2012,27(1):117-122.
[3]HUANG Z M,XIE Z S,HOU C M.On a Discrete Fractional Boundary Value Problem with Nonlocal Fractional Boundary Conditions[J].Chin,Quart,J,of Math,2014,29(4):539-552.
[4]LI Y,JIANG W,HU B B.Stability of Neutral Fractional Neural Networks with Delay[J].Chin,Quart,J,of Math,2016,31(4):422-429.
[5]LI X Y,XIANG J R,WU Y Y.Laplace Transform Method Applied to Solve Fractional Fractional Equations[J].Chin Quart J of Math,2015,30(1):121-129.
[6]MATIGNON D.Stability results for fractional differential equations with applications to control processing,in:IMACS-SMCProceedings,Lille,France,1996.
[7]GABANO J D,POINOT T.Fractional modelling and identification of thermal systems[J].Signal Process,2011,91:531-541.
[8]LIN J,POINOT T,TRIGEASSOU JC,KABBAJ H,FAUCHER J.Modelisation et identification d,ordre non entire d,une machine asynchrone.In:Conference Internationale Francophone d,Automatique 2000.
[9]LI Y,CHEN Y,PODLUBNY I.Stability of fractional-order nonlinear dynamic systems:Lyapunov direct method and generalized Mittag Leffer stability[J].Comput Math Appl,2010,59:1810-1821.
[10]MANUEL A,DUARTE M,NORELYS A C.Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems[J].Commun Nonlinear Sci Numer Simulat,2015,22:650-659.
[11]NORELYS A C,MANUEL A,DUARTE M.Lyapunov functions for fractional order systems.Commun Nonlinear Sci Numer Simulat,2014,19:2951-2957.
[12]ANAYA J F,ANTONIO J N,GALANTE J J.Lyapunov functions for a class of nonlinear systems using Caputo derivative[J].Commun Nonlinear Sci Numer Simulat,2017,43:91-99.
[13]JAVIER A,GALLEGOS M A,DUARTE M.On the Lyapunov Theory for fractional order systems[J].Applied Mathematics and Computation,2016,287:161-170.
[14]LI C P,ZHANG F R.A survey on the stability of fractional differential equations[J].European Physical Journal Special Topics,2011,193:27-47.
[15]SONG L,WU R C,CHEN L P.Comparison principles and stability of nonlinear fractional-order cellular neural networks with multiple time delays.Neurocomputing,2015,168:618-625.
[16]ZHANG L,SONG Q K,ZHAO Z J.Stability analysis of fractional-order complex-valued neural networks with both leakage and discrete delays.Applied Mathematics and Computation,2017,298:296-309.
[17]HU J B,ZHAO L D,LU G P,ZHANG S B.The stability and control of fractional nonlinear system with distributed time delay,Applied Mathematical Modelling,2016,40:3257-3263.
[18]SABATIER J,MOZE M,FARGES C.LMI stability conditions for fractional order systems[J].Computers and Mathematics with Applications,2010,59:1594-1609.
[19]GUO B L.Fractional Partial Differential Equations and their Numerical Solution[M].Science Press of China,2011.
[20]KILBAS A,SRIVASTAVE H,TRUJILLO J.Theory and Applications of Fractional Differential Equations[M].Elsevier,North Holland 2006.
[21]PODLUBNY I.Fractional Differential Equations[M].Technical University of Kosice,Slovak Republic,1999.