Characteristics Of Solutions Of Interval Systems Over Max-plus Algebras

In engineering field,the problem of solving uncertain systems has always been a theory problem.The problem mainly comes from two aspects:coefficient matrix and uncertainty of variables.Researchers have been trying to break through this uncertainty,so that the reconstructed mathematical model has a relatively robust effect.In the past few decades,research scholars try to solve this uncertain system by calculus.In recent years,scholars have proposed a new method for solving uncertain systems by introducing intervals into uncertain systems,the problem of solving uncertain systems can be solved by exploring the characteristics of interval solutions.After introducing the concept of interval,we call the mathematical model of the uncertain system interval system.Researchers understand the characteristics of various problems of this kind of interval system to reflect the mathematical model of uncertainty and its stability.With interval systems being used as a breakthrough to solve uncertain problems,more and more scholars begin to pay attention to this new field.In recent years,the study of the characteristics of various solutions under the maximal algebraic max-plus structure is also in full swing.In this thesis,based on the basic concepts and application background of max-plus structure of maximal algebra,we mainly study some problems about the characteristics of the solutions of related systems under the max-plus structure of maximal algebra.The first chapter is the introduction.Firstly,the background and significance of the research on the characteristics of the solutions of the related systems under the max-plus structure are introduced in detail.Secondly,the basic theory of intervals and symbolic explanations related to this paper are briefly summarized.Finally,the research status of the characteristics of the solutions of the related systems under the max-plus structure is briefly summarized.In the second chapter,we discuss the related definitions and necessary and sufficient conditions of the control solution for the interval equations of A(?)x=b.Myskova H.et al.gave only the necessary and sufficient conditions for the weak,strong and tolerance solutions of interval equations of A(?)x = b.In order to improve the characteristics of solutions of interval equations of A(?)x=b.First of all,the definition and description of the control solution of interval equations of A(?)x=b are presented.Then,in order to get the relevant conclusions of this chapter,by studying the max-plus structure of maximal algebra,we prove that the transformation of coefficient matrices under this structure is order-preserving.By using the order preserving lemma,the necessary and sufficient conditions for the corresponding control solutions of the interval equations of A(?)x=b are given.On the basis of theoretical proof,corresponding examples are given to depict the relevant conclusions drawn in this chapter.In the third chapter,we discuss the weak,strong,tolerance and control solu-tions of the system of interval bilateral inequalities under the structure of max-plus algebras.Firstly,by defining weak solutions,strong solutions,tolerance solutions and control solutions of interval bilateral inequalities under the structure of max-plus algebra,the necessary and sufficient conditions for weak solutions,strong solution-s,tolerance solutions and control solutions of interval bilateral inequalities under max-plus algebra are established.Second,we have a preliminary understanding of the properties of various solutions under the structure of max-plus algebras,and then provide a theoretical basis for the optimization problems under the structure of max-plus algebras.In the fourth chapter,we discuss the AE solution of two sided interval equa-tions and interval inequalities.Firstly,the concepts of AE solutions of two-sided interval equations and interval inequalities are introduced in the max-plus structure of maximal algebra.Secondly,the necessary and sufficient conditions of AE solu-tions on the max-plus structure of maximal algebra are established on the basis of the conservative lemma in the first chapter.Finally,it is deduced that the necessary and sufficient conditions for various solutions(weak solutions,strong solutions,tol-erance solutions and control solutions)on the max-plus structure are a special case of the AE solution,and the main conclusions are verified by several examples.The fifth chapter summarizes the main research contents of this thesis,and on this basis,the future research direction and research content are prospected.