Qualitative Analysis Of Solutions To Second-order Impulsive Differential Equation

Second-order impulsive differential equations have a wide range of applications in biology,engineering,economics and other fields.Its qualitative analysis is an important guide for understanding natural phenomena,the operation of the project and solving economic problems.In this paper,we start from the second-order linear impulsive differential equation,and give the explicit expression of the solution of the linear problem by using the series expansion of the uniformly continuous cosine operator family,then apply it to the nonlinear problem and its control problem.The study focuses on the existence,continuous dependence,differentiability,stability,periodic solutions,and controllability of the solutions of nonlinear problem.The details of the dissertation are as follows:Firstly,the stability of first-order impulsive differential equation is extended to second-order impulsive differential equation.For linear problem,with the help of the expression for the solution of the linear problem,we derive some estimates of the solution by describing the relationship between the spectral radius of the matrix,the pulse point and the length of the interval,and then we give some sufficient conditions for the asymptotic stability of the zero solution.For the perturbation problem of linear equations,a number of sufficient conditions are given to ensure the exponential stability of the solution of the perturbation problem under the condition that the perturbation matrix is uniformly bounded.For nonlinear problem,the existence and uniqueness of solution is proved by using the fixed point theorem,and the stability of Ulam-Hyers-Rassias is further investigated.Secondly,the idea of proving the continuous dependence and differentiability of the solution of the ordinary differential equation with respect to the initial value is transferred to the second-order impulsive differential equation.For the problems of simultaneous perturbation of the initial value and the impulsive points,the existence and uniqueness of the solution is given by constructing an iterative sequence of Picard.On this basis,the approximation relation between the perturbed problem and the original problem is given,which is used to derive the continuous dependence and differentiability of solutions with respect to initial value and impulsive points.Thirdly,we develop the method of classical impulsive period problem,and study the existence and uniqueness of periodic solutions to the second-order impulsive differential equation.By constructing an appropriate concomitant system,we give the necessary and sufficient condition for the existence and uniqueness of periodic solutions for the linear problem.For the nonlinear problem,by constructing a -compact operator and a Fredholm operator with zero index,the existence problem of periodic solution of the original problem is transformed into a solution problem of corresponding operator equation,and then the existence and uniqueness of the periodic solution is given by using the fixed point theorem and the coincidence degree theory.Finally,on the basis of qualitative analysis,the controllability of the solution of the Cauchy problem for the second-order impulsive differential equation is studied.The Gramian matrix criterion and the rank criterion are given for the linear problem.For the nonlinear problem,the controllability problem is transformed into the existence problem of the fixed points of the operator equation by constructing appropriate control functions,and then the controllability results are obtained.