Qualitative Theory Of Several Classes Of Delay Differential Systems

Delay differential equations can more accurately describe the change law of objective things than ordinary differential equations,and have many applications in the fields of biology,engineering,economics and so on.However,different from ordinary differential equations,delay differential equations are essentially infinite dimensional systems.Even the corresponding characteristic equation of linear delay differential equation is transcendental equation,which makes it difficult to give an accurate representation of the solution,and then affects the qualitative analysis of the system.Therefore,it is very important to give an accurate representation of the solutions of delay differential systems,which can also lay a foundation for the study of system stability and controllability.In this paper,the constant variation method in ordinary differential equations is extended to delay differential systems.By constructing the appropriate delayed matrix exponential,the exact solutions of several classes of delay differential systems are obtained.On this basis,the stability and relative controllability of the solution are studied.Firstly,as the basis of subsequent research,nonsingular delay differential systems are studied.By constructing the delay matrix exponential of linear systems,the explicit solutions of homogeneous and non-homogeneous problems are derived.Moreover,sufficient conditions for the exponential stability of the solutions of the system are given.By introducing the delay Gram matrix,the sufficient conditions for the relative controllability of linear systems and semilinear systems are given respectively.Secondly,Caputo type fractional delay differential systems and neutral delay differential systems are studied.The fractional delay Gram matrix and neutral delay Gram matrix corresponding to linear systems are constructed respectively,and the Gram criteria for controllability of two kinds of linear systems are given.For Semilinear problems,sufficient conditions for relative controllability of the systems are given by using Krasnoselskii's fixed point theorem.Thirdly,the impulsive single delay differential system is studied.Firstly,the impulsive delayed matrix exponential is constructed and its properties are analyzed.The explicit solution of linear impulsive single delay differential system is given by the impulsive single delay matrix exponential,the constant variation method and the superposition principle.Next,with the help of the expression of the solution of Semilinear impulsive single delay differential system and the corresponding Gronwall inequality,some sufficient conditions for the asymptotic stability and finite time stability of impulsive single delay differential systems are given when the semilinear term satisfies different conditions.Finally,the Gram criterion for linear impulsive single delay control systems is given,and the sufficient conditions for the relative controllability of Semilinear systems are given by using the fixed point theorem.Finally,the research method of impulsive single delay differential system is extended to impulsive multi-delay differential system.The impulsive multi-delayed matrix exponential is successfully constructed,and then the expression of the solution of impulsive multi-delay differential system is given,which is applied to the analysis of system stability and relative controllability.Due to the interaction of multiple impulsive points and multiple delays,the research difficulty is increased.This paper lays a necessary theoretical foundation for the further study of the optimal control problem of time-delay systems.