Several Properties&Optimal Control Of Riemann-Liouville Uncertain Fractional-order Difference Equations

Fractional-order difference(differential)equations can describe the systems with memory and hereditary feature more accurately,and the systems usually work with uncertainty.When the possibility of the event happening with uncertainty can not be approximated by frequency,we may use belief degree to approximate it.Uncertainty theory is an effective axiomatic system to deal with belief degree.The optimal control problem is widely existed in various fields.However,research on fractional-order difference equation and optimal control in this uncertain setting is not perfect.Based on uncertainty theory,this thesis will define fractional-order uncertain difference equations and explore the related properties and optimal control problem of fractional-order uncertain difference equation for Riemann-Liouville(R-L)type.The main work is as follows:1.The solution and its existence are studied for R-L type fractional-order uncertain difference equation.(1)The analytic solutions to two classes of the proposed equations are derived by Picard successive approximation method;(2)Using Banach fixed-point theorem,we derive that there is almost surely a unique solution to the proposed equations whose coefficients satisfy Lipschitz conditions with bounded Lipschitz constants.2.The uncertainty distribution function of the solution is investigated for R-L type fractionalorder uncertain forward difference equation.By comparison theorem and α-path,it is proved that the inverse uncertainty distribution function of the solution to the proposed equation is itsα-path.3.The finite-time stability is researched for R-L type fractional-order uncertain backward difference equation.(1)The finite-time stability in mean and in measure are defined for the proposed equations,respectively;(2)According to the definitions and the special inequalities,the sufficient conditions of finite-time stability in mean and in measure are derived,respectively.4.A necessary condition of the optimal control is investigated for controlled systems described by a class of R-L type fractional-order uncertain difference equations.The necessary condition of optimal control obtained by using the classical variational method is to solve a system of R-L type fractional-order uncertain backward difference equations.5.Linear quadratic optimal control problems are considered for controlled systems described by two classes of R-L type linear fractional-order uncertain difference equations.(1)A linear quadratic optimal control problem for a class of the proposed systems is solved by the necessary condition of optimal control obtained;(2)A linear quadratic optimal control problem for the other one is solved by dynamic programming method,and the optimal control is a linear combination of the current state and the past states.