Performance Analysis And Applications For Uncertain Fractional Dynamical Systems

Fractional calculus is widely used to describe the memory and hereditary properties of real matters,due to the fact that it introduces convolution integrals with power-law memory kernels.In recent years,fractional calculus has achieved many successful practices in the fields of anomalous diffusion,system control and economics,etc.,which also prompts fractional differential equation to play an irreplaceable role in depicting the global correlation and historical dependence characteristics contained in the dynamic evolution process of the system.However,indeterminate factors are widely existing in the evolution of systems.An indeterminate factor of the system can be described by a dynamical model based on stochastic process when there is sufficient sample to reflect it.Nevertheless,in many engineering fields,it is difficult to obtain the required samples and accurate observation data through plentiful repeated experiments due to the limitations of funds and technology.We can only obtain the reliability of the likelihood of an indeterminate event based on empirical knowledge of domain experts,which makes the description of the system often accompanied by uncertainty caused by human subjective preference.In this case,uncertain fractional differential equations based on uncertainty theory can be used to model the evolution process of the systems.The nonlocality of fractional order operators shows the evolution law of the system precisely,but also brings many difficulties in studying the performance of uncertain fractional dynamical system.In this paper,we study the properties,the uncertainty distribution of solution,impulsive problems and real applications,etc,for a class of uncertain fractional dynamical systems,the details of main work are proposed as follows.1.Firstly,the general definition of a class of uncertain fractional differential equations is given,and the analytic solution for a linear uncertain fractional Cauchy problem is obtained with the help of Mittag-Leffler functions.Next,the continuity in measure sense of the solution to uncertain fractional Cauchy problem with respect to the initial points and parameters are studied,respectively,based on the generalized Gronwall inequality.Moreover,the stability in measure of uncertain fractional dynamical systems is defined and the sufficient conditions are derived for ensuring the stability of these systems of order 1/2<p?1 and 0<p?1/2,respectively,by employing inequality technique.2.The uncertainty distribution of solution for initial value problems of a class of uncertain fractional differential equations is studied.In the first place,comparison theorems for Caputo fractional Cauchy problem are improved and proved strictly.Then the definition of ?-path for an initial value problem of uncertain fractional differential equation is introduced,which is derived to be the inverse uncertainty distribution of the solution to this problem based on the proposed comparison principles.The numerical algorithms of calculating the inverse uncertainty distribution of the solution and the expected value of a monotone function with respect to the solution are constructed,respectively,by employing fractional Adams method.Finally,the effectiveness and accuracy of the algorithm are analyzed by several numerical examples.3.The solution of impulsive problem for a class of uncertain fractional order systems is investigated.Firstly,the concept of impulsive uncertain fractional differential equation is proposed with the integral equation of its solution,and analytic solutions of two linear uncertain impulsive problems are obtained,respectively.Then the sufficient conditions for the existence and uniqueness of the solution to uncertain fractional impulsive problem are derived based on the Kuratowski non-compact measure and related fixed point theorems,respectively.Finally,an illustrative example is provided to explain the proposed results.4.The Asian option pricing problem is studied based on uncertain fractional differential equation.A stock model with mean reversion is established by the Caputo type of uncertain fractional differential equation.The pricing formulas of the Asian options are derived under expected value and optimistic value criteria.And some numerical experiments are provided to illustrate the economic meanings of the obtained results.