Periodic-Integral Boundary Value Problems Across Resonance

A differential equation (DE in short)is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. Ordinary differential equations (ODE in short)servc as mathematical models for many exciting practical problems, not only in science and technology, but also in such di-verse fields as economics, psychology, defense, and demography. With the rapid growth in the theory of itself and the huge applications to almost every branch of knowledge DEs play a prominent role in our daily life moreover it has resulted in a continued interest in its study by students in many disci-plines.Boundary value problems of differential equations or ordinary differential equations (BVP in short)arise in many areas of science and technology. In this thesis, we make attempt to use the Optimal Control Theory to solve ODE's BVPs. As we know the theory of control of differential equations has developed in several directions in close relation with the practical applications of the theory. Its evolution has shown that its methods and tools are drawn from a large spectrum of mathematical branches such as ordinary differential equations, real analysis, functional analysis, calculus of variations, mechanics, geometry. Without being exhaustive we just mention, as subbranches of the control theory, the controllability, the stability, the observability, the optimization of differential systems and of stochastic equations or optimal control. With the above in mind we briefly in chapter 1, section 1-2, our goal is to give a brief introduction to the ODE's BVP and the Optimal Control including the history, the typical problems, the techniques, etc. In Chapter 1, section 3, we start by introduced some important results have lots of closed connection to our issue. In view of the indicated relationship with problems in the section, many obtained results can be also employed for solving ODEs' BVPs and we are not to be too shocked by discovering that by using topological principles such as the fixed point method and the homotopic method, huge amounts of ODE's problems could be handled. The last section in chapter 1, by tracing the development of our problem from its earliest general results to what may be considered as the most recent results so far obtained, we will give a conclusion to the past and now on our thesis. The whole Chapter 2, as well as a preview of the knowledge on ODE. optimal control, nonlinear analysis, and techniques, concepts and examples or so what we must be at least familiar with.Inspired by the series works of Wang and Li on ODE's BYPs [92.93, 138.139,140,141], we arc on the stage to continue the research on an ODE's BVP, which is never been touched specifically to those problems having significant application to the human life, named periodic-integral bound-ary value problem. Via to the one of the theories of the optimal control, Pontryagin Maximum Principle, we can deal with the unique solvability for the periodic-integral boundary value problems for second order differential equations across resonance. Chapter Three and Four are the main parts in the text. Before presenting the main theories of our issue we deal with the associated linear systems and we can easily prove the existence of a unique solution of the control systems, respectively. According to the definitions and properties we mentioned before, the optimal condition of the unique solvability to the periodic-integral boundary value problems for second order differential equations are given by serious analysis.The results are extended to the nonlinear systems and we will give the proof in the end.Optimal Solvability for Periodic-integral Boundary Value Problems across one resonantFirstly we consider the following periodic-integral boundary value prob-lems for second order differential equations across one resonant.together with the assumptions as follows:(H1) f(t,y),fy(t,y)are continuous on[0.2π]×R1;(H2) 0