Research On Stability Of Fractional-order Nonlinear Systems

Fractional calculus unifies and generalizes the integer-order calculus. As anapplication background of fractional calculus, fractional-order systems have receivedincreasing research attention. It not only extends the classical integer-order systemtheory, but also provides an excellent tool for the description of various processes.Recently, fractional calculus was introduced to the stability analysis of nonlinearsystems. Although there are some results on stability of fractional-order nonlinearsystems so far, the number is very limited. It should be noted that it is difficult toevaluate the stability for fractional order dynamic systems by simply examining itscharacteristic equation either by finding its dominant roots or by using other algebraicmethods. In addition, it is well known that Lyapunov direct method cannot be simplyextended and applied to the case of fractional-order, although many stability resultsabout integer-order systems are obtained by constructing a suitable Lyapunovfunctional. Stability analysis of fractional-order nonlinear systems is a very dauntingtask.The problem of stability is a very fundamental and crucial issue forfractional-order neural networks, however, due to the high complexity of fractionalcalculus, it has been investigated and discussed only in some recent literature, and onlyvery few relevant results have been obtained, stability analysis of fractional-orderneural networks is urgent and meaningful.Based on the current status of the research, the main contribution of this paper isas follows:Firstly, the stability of a class of Caputo fractional-order nonlinear systems withfractional-order：0<1is discussed. By using the property of fractional calculusand Gronwall inequality, stability theorem for such fractional-order nonlinear systemsis proven. Based on the stability theorem, coresponding linear state feedback controlleris designed to stabilize such fractional order nonlinear systems. A numerical exampleis given to illustrate the effectiveness of the proposed method.Secondly, the stability of a class of Caputo fractional-order nonlinear systemswith fractional-order：1<2is discussed. By using the property of fractionalcalculus and Gronwall inequality, stability theorem for such fractional-order nonlinearsystems is proven. Based on the stability theorem, coresponding linear state feedback controller is designed to stabilize such fractional order nonlinear systems. A numericalexample is given to show the validity of the proposed method.Thirdly, the problems of the stability and existence and uniqueness of solutionsfor a class of fractional-order neural networks are studied by using Banach fixed pointprinciple and analysis technique. A sufficient condition is given to ensure the uniformstability of solutions and existence and uniqueness of solutions for fractional-orderneural networks with variable coefficients and multiple time delays.Finally, combined with Banach fixed point principle and analysis technique,uniform stability of solutions and existence and uniqueness of solutions are investigatedfor a class of fractional-order non-autonomous systems with multiple time delays.