Time Optimal Control Problem For A Class Of Linear Ordinary Differential Equations

In this thesis,the time optimal control problem of a class of linear ordinary differential equations in R~n is considered,and the problem's control constraint set is a rectangular control constraint set.First of all,the equivalence theorem between the time optimal control problem and norm optimal control problem is established.Then the sufficient and necessary condition for the optimal time and time optimal control is obtained.Finally,a numerical method is presented for the computation of the optimal time and time optimal control.In the existing conclusions,the target set of the linear ordinary differential equations is the origin.However,the target set of the control system considered in this thesis is a closed ball containing the origin.The thesis consists of five chapters.In chapter 1,the research background and research content are introduced.First of all,the origin and development of the control discipline is reviewed.Then the time optimal control problem and norm optimal control problem are introduced.Moreover,the research and application of the equivalence theorem are reviewed.The chapter 2 is preparatory knowledge.First of all,the time optimal control problem and norm optimal control problem of a class of linear ordinary differential equations in R~n are introduced.The target set of the control system is a closed sphere containing the origin,and the control constraint set is a rectangular control constraint set.Furthermore,the basic hypothesis,the definition of basic symbol and lemmas are introduced.In chapter 3,the equivalence theorem between the time optimal control problem and norm optimal control problem for a class of linear ordinary differential equations is proved.Moreover,the optimal time function and the optimal norm function are continuous and strictly mono-tone decreasing functions.In chapter 4,the sufficient and necessary condition for the optimal time and time optimal control is obtained.The proof technique is to construct the appropriate functional so that the minimizer can be used to describe the optimal control of the norm optimal control problem.Moreover,the main conclusion of this chapter is obtained by combining the equivalence theorem.In chapter 5,a numerical method is presented for the computation of the optimal time and time optimal control,and some examples are given to illustrate the effectiveness of this method.