Logical Matrix Equation And Its Applications In Boolean Networks

Using semi-tensor product of matrices,Boolean networks,including Boolean control networks and switched Boolean networks,can be converted into their equivalent algebraic representations.Based on their new forms,numerous issues in Boolean networks can be solved systematically.To deal with some problems on design,a method of logical matrix equation is proposed.Under this constructive approach,coordinate transformations in block decomposi-tion,inverse Boolean control networks in nonsingularity analysis,switching signals to achieve stabilizability,etc.,can be designed.Chapter 1 of this dissertation introduces the origin and developments of matrix equations and Boolean networks,especially in the aspects of control-lability and observability,decomposition and decoupling,invertibility,stabi-lizability and so on,via semi-tensor product of matrices.Chapter 2 presents some necessary preliminaries on semi-tensor prod-uct and shows the process to transform Boolean control networks and switched Boolean networks into equivalent algebraic representations in detail.In Chapter 3,canonical solutions sets of three kinds of logical matrix equation,which are derived from practical problems,are obtained.Moreover,special form of logical matrices makes it possible that we can all required canonical solutions in a concise method.These results will be applied into subsequent chapters to decompose systems,to compute inverse systems,and to design switching signals.In Chapter 4,the block decomposition of Boolean control networks is investigated via solving logical matrix equations.Firstly,the definition of block decomposition of Boolean control networks is proposed.Secondly,the block decomposition problem is equivalently converted into the solvability of a set of logical matrix equations,based on which the suitable coordinate transformations can be determined.Finally,an illustrative example is given to show the effectiveness of obtained results.Chapter 5 devotes to the input observability,which also is called the nonsingularity or left invertibility,of general Boolean control networks.First,using graphic approach converts the global nonsingularity into pointwise finite-step nonsingularity.Subsequently,a series of matrix-form criterions are given to identify the nonsingularity,under which suitable inverse Boolean control networks can be designed.Finally,one numerical example are used to illustrate the feasibility and validity of the obtained results.In Chapter 6,three kinds of stabilizability of switched Boolean net-works,which are stabilizability under arbitrary switching signals,pointwise stabilizability and consistent stabilizability,are analyzed.Under the frame-work of an improved approach,stabilizability under arbitrary switching sig-nals and pointwise stabilizability are discussed,respectively.On the basis of adjoint logical networks,which are constructed from the considered switched Boolean networks via the types of switching signals,consistent stabilizabil-ity and stabilizability under time-varying output feedback axe investigated.Some matrix criterions are established to determine stabilizability.Com-pared with existing achievements,results in this chapter have less complexi-xii ty.Chapter 7 further investigates stabilizable switching signals design of switched Boolean networks via semi-tensor product of matrices.First,a con-structive method is presented to determine all stabilizable switching signals dependent on states,which drive the considered plant stabilizable to a fixed point.Then,globally consistent stabilizability of switched Boolean network-s is equivalently converted into local(one state)pointwise stabilizability in adjoint logical networks,under which all consistent switching signals in the sense of the finite-time stabilizability can be designed.Then the method is generalized to design stabilizable time-varying output feedback switching signals.Moreover,the feedback capability of switched Boolean networks is discussed via logical matrix equations.Finally,a biological example and a digital electronic system is discussed to show the effectiveness of obtained results.Chapter 8 concludes this dissertation,including defects existed in this study and prospects of Boolean networks.