Study On Stability And Controllability Of Conformable Fractional Differential Systems With Delay

Compared with the classical theory of fractional calculus,conformable fractional calculus has many advantages in form and property,and it has achieved rapid development and wide application in theoretical analysis and practical engineering in recent years.However,as a common physical phenomenon in the objective world and the practical systems,the time-delay effect has not been fully mentioned and considered in the existing research works related to this well-behaved fractional calculus.With this as the motivation,this paper mainly discuss the solutions of delay differential systems and their behavior problems under the structure of conformable fractional derivative,including the existence theorems of solutions for initial value problems of delay differential equations,the finite-time stability,Ulam's type stability and the controllability of delay differential systems and so on.The research content of this paper is mainly divided into the following four parts:In the first part,by utilizing the technique of conformable fractional Laplace transform,we investigate the Ulam-Hyers stability and the Ulam-Hyers-Rassias stability for several kinds of linear conformable fractional differential equations.At the beginning in the second part,some fixed point theorems are used to analyze the existence problems for the initial value problems of nonlinear conformable fractional functional differential equations.Then,we use the Gronwall inequality in the sense of conformable fractional integral to establish the existence and uniqueness theorem of solutions and the finite-time stability criteria of a class of nonlinear conformable fractional delayed differential systems,the estimate inequality of solution is given at the same time.In the third part,we first discuss the Ulam-Hyers stability and the generalized Ulam-Hyers' type exponential stability of differential-difference equations in the frame of conformable derivative via different methods including the Gronwall inequality and Picard operator,then we utilize the iterated integral inequality and the approach of Picard operator to study the Ulam-Hyers stability of nonlinear Volterra delay integro-differential equation.In the last part,by establishing an appropriate admissible control function,the controllability problem of the system is transformed into the fixed point problem of the corresponding operator in the function space.The controllability problems of semilinear conformable functional differential systems with infinite delay in Banach space under traditional and non-local initial conditions are studied respectively.