Topological Structure And Optimal Control Of Singular Mix-valued Logical Networks

With the development of biological and genetic systems, mix-valued logi-cal networks have played an important part in both theoretical and practical fields. As a generalization of singular Boolean networks, singular mix-valued logical networks have been applied in many aspects. This paper introduces singular mix-valued logical networks and singular mix-valued logical control networks. By semi-tensor product, the complex logical networks can be con-verted to the algebraic form. Based on that, we discuss the normalization problem and solvability. Also, the necessary and sufficient condition for the existence and uniqueness of solutions is presented. Then, concepts of fixed points and cycles are also studied. Furthermore, two kinds of optimal control of singular mix-valued logical control networks, including similar Mayer-type optimal control and minimum-time control, are presented. The sufficient con-dition for the existence of the optimal control is supplied respectively and the solving methods are provided. At last, illustrative examples are given to show the feasibility of the results. We organize this thesis as follows:In Chapter 1, the research background of singular systems and singular mix-valued logical networks are provided. Also the research status of singu-lar Boolean networks and singular mix-valued logical networks are reviewed. Moreover, the semi-tensor product of matrices is discussed. At last, the main research problem and organization of this thesis are presented.Chapter 2 introduces the theory of semi-tensor product and several neces-sary results about it. Furthermore, the logical operator, logical functions and mix-valued logical networks are discussed. Based on the definition of structure matrix, the mix-valued logical networks can be converted into algebraic form.In Chapter 3, some prior knowledge of dynamic-algebraic mix-valued log-ical networks is provided. Then its normalization and solvable problem are studied. The equivalent form for the existence and uniqueness of solution is supplied. Finally, the fixed points and cycles are discussed.Two kinds of optimal control problems of singular mix-valued control problems are investigated in Chapter 4. Firstly, singular mix-valued control networks are converted into algebraic form. When it has the unique solution, the network can be transformed to an equivalent form, based on which the Mayer-type optimal control problem and minimum-time control problem are discussed.Finally, Chapter 5 concludes this thesis.