The Study On Two Classes Of Nonlinear Impulsive Differential Equations(System) And Their Optimal Control

In this paper, the existence and uniqueness of positive solutions for two classes of non-linear impulsive differential equations (system) and their optimal controls are discussed by using the cone and partial order theory as well as the properties of operators. The new results extend and improve some findings in the relevant literatures. The overall structure of this paper is as follows:In Chapter1, the background of the discussed problems and the necessity of our study in this paper is introduced briefly. Meanwhile, the main works of this paper are stated in detail.In Chapter2, we study a coupled system of second order impulsive differential equations and its optimal control. Mainly by using the proper construction of an auxiliary operator and a fixed point theorem of concave operators, we solve the problem with coupled variables and obtain the existence and uniqueness of positive solutions as well as the existence of an optimal control.In Chapter3, we study an initial value problem of a fourth order impulsive differential equations with mixed monotonicity and its optimal control. We solve the problem with variable coefficients and mixed monotonicity in the nonlinearity and get the existence and uniqueness of positive solutions as well as the existence of an optimal control by using a fixed point theorem of mixed monotone operators. What’s more, we consider the stability of the optimal control.In Chapter4, combining with the background of application in practical, we discuss a boundary value problem of fourth order impulsive differential equations with mixed mono-tonicity: Mainly by using the cone theory and fixed point theory, we obtain the existence and uniqueness of positive solutions to our problem. Especially, the method discussed in this chapter doesn’t require the condition that the nonlinearity to be continuous and is suitable for the general problem of elastic beam equations. Also it improves the result when an elastic beam subjected to a nonlinear foundation. Lastly, as an application, we give related examples to illustrate our results.