Studies on Stability and Applications for Fractional Differential Systems

Author:Yan Ye

Supervisor:Kou Chunhai


Degree Year:2012





Fractional calculus may be considered an old and novel topic.It is an old topic since the origin of fractional derivative and integral can be traced back to the end of seventeenth cen-tury,that is,it almost has the same long history as that of the conventional calculus.However,,it may be considered a novel topic as well,since,in the ensuing nearly three centuries,it is studied basically in the field of pure mathematics.Though a list of mathematicians,who have provided important contributions up to the middle of the last century,includes Euler(1730),Laplace(1812),Fourier(1822),Liouville(1842),Riemann(1876),Riesz(1949),the frac-tional calculus is only a subject in which mathematicians are interested.Because there are several kinds of different definitions proposed for fractional derivative and integral and the researchers have not found the physical background and geometric meaning of fractional cal-culus,the study of this field develops very slowly.However,in the past forty years,a lot of research and experiments show that fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of materials and processes.This is the main advantage of fractional derivatives in comparison with classical integer-order models,in which such effects are in fact neglected.With more and more applications to science and en-gineering,many fractional differential models have appeared and it has attracted much more attention of researchers in theory and applications.This dissertation concerns about stability analysis and applications for fractional differ-ential systems.It is divided into seven chapters.In Chapter One,we briefly introduce the history of fractional calculus and fractional dif-ferential equations.We overview the progress in the theory of fractional differential systems and put emphasis on their applications in control,viscoelastic mechanics,etc..Then we make an exposition of known results on stability for fractional differential systems.Finally,we introduce several definitions of fractional calculus used in this paper.Chapter Two is contributed to study for the global existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis,which is fundamental in the basic theory of fractional differential equations and also important in stability analysis of this kind of equations.A special Banach space is introduced and an appropriate compactness criterion is established,such that we can use the Leray-Schauder alternative theorem to obtain some sufficient conditions which guarantee the global existence of solutions on the half-axis.By considering a more general space,some further results are obtained.We also present an example to support our results.The practical stability for fractional impulsive hybrid systems is discussed in Chapter Three.The transformation from a linear fractional differential system to a fractional impulsive hybrid system is interpreted;with the help of Mittag-Lefler functions of matrix-type,the explicit solution of a fractional differential system is obtained.Then we derive some sufficient conditions which guarantee the practical stability for the systems that we study.And finally,an example is given to illustrate the validity of our results.In Chapter Four,we firstly use the linear approximation method to discuss the stability properties for fractional order differential equations,and obtain an sufficient condition which guarantees the system asymptotically stable.Then we introduce the fractional derivative to a model of HIV-1 infection.This model has three equilibria:the trivial equilibrium,the healthy equilibrium and the infected equilibrium.We consider the asymptotical stability for the healthy equilibrium and the infected equilibrium,and some simulations are presented to illustrate the validity of the results.In Chapter Five,a fractional-order model of HIV infection with time delay is proposed,where the time delay is introduced to describe the time between infection of the health cell and the emission of viral particles on a cellular level.In our analysis,there are two positive equilibria:the uninfected equilibrium and the infected equilibrium.First,we consider the asymptotical stability properties of the uninfected equilibrium.Moreover,we point out an improper result in others’ paper.Then,we consider the stability properties of the infected equilibrium.We get that the infected steady point is stable despite of the size of the delay under certain conditions.Subsequently,in Chapter Six,we introduce another fractional model of HIV-1 infection with time delay.In this model,there are two positive equilibria:the uninfected equilibrium and the infected equilibrium.First,we consider the asymptotical stability properties of the uninfected equilibrium.We get a sufficient condition on the parameters for the stability of the uninfected steady point.Next,we consider the stability properties of the infected equilibrium.By analyzing the corresponding transcendental characteristic equation,we analytically derive some asymptotical stability conditions for the infected steady state.Finally,a summary of the thesis is made,and the further development is put forward.