Dynamics And Control Of Boolean Networks And Finite-Field Networks

In actual biological networks(multi-agent systems),genes(agents)are generally equipped with limited memory,computation,and communication capabilities.Hence,it is reasonable to assume that each gene(agent)is capable of storing,processing,and transmitting exclusively elements from a finite and pre-specified alphabet.In this setup,gene regulatory networks are described as Boolean networks,where the level of each gene is quantized into two states(1 and 0),and the state update of each gene is assigned by a certain logical rule.Besides,in the framework of multi-agent systems over finite fields,the state of each agent is assigned from a finite alphabet,and the operations are performed according to modular arithmetic.Inspired by the superiorities of Boolean networks and finite-field networks,this thesis dedicates to studying their dynamics analysis and control design.The main works are as follows:The background knowledge of Boolean networks and finite-field networks,as well as the current research status are introduced.Then,the relations and differences between Boolean networks and finite-field networks are explained.After that,the main methods and theories used in this paper are presented,including the semi-tensor product of matrices,algebraic state space repression,the concept of finite fields,and algebraic graph theory.The main content is divided into two parts.The first part focuses on Boolean networks,and investigates the dynamics analysis and control design in the situations of state constraints,impulsive effects and probabilistic switching.With respect to the constrained Boolean networks,the topological structure,constrained reachability and constrained stabilization under event-triggered control approach are investigated,and the concepts of constrained fixed point and constrained cycle are proposed,where constrained fixed point has two types:livelock one and deadlock one.Subsequently,a formula is given to calculate the number of constrained fixed points and constrained cycles,and an event-triggered control strategy is proposed to stabilize constrained Boolean networks.As for impulsive Boolean networks,in order to characterize the detailed impulsive process,a hybrid index model is established for impulsive Boolean networks via sampled-data control.Based on this model,the algorithms for finding the largest sampled point control invariant set in the hybrid domain and time domain are designed,and the checking criteria for the set stabilization of sampled-data-controlled impulsive Boolean networks are proposed.These results can also be applied to solve several problems,including synchronization,output tracking,and stabilization to certain states.It should be pointed out that the methods used in the existing works related to Boolean networks are almost based on the algebraic state space representation,which may lead to a high computational complexity as the number of nodes increases.Motivated by such drawback,a novel pinning control strategy based on the network structure is explored,which overcomes the limitation of control design based on the traditional algebraic state space representation,and avoids the problem of constructing the high-dimensional state transition matrix.In particular,this novel pinning control method is applicable to large-scale probabilistic Boolean networks with spare connections,which happen to be the case in biological networks.The second part of the main content focuses on finite-field networks,and dedicates to investigate the synchronization of stochastic finite-field networks and the impulsive control design for finite-field networks.During the research process,one discovers that finite-field networks have several interesting phenomena that are different from real-valued networks and logical networks.For example,the finite-field synchronization in the stochastic sense has the relations that fixed-time synchronization with probability one is equivalent to finitetime synchronization with probability one,and it can deduce each mode being synchronous;besides,the synchronization of each mode deduces the asymptotic synchronization with probability one of the whole network.Since open issues remain in control design for finitefield networks,the impulsive control is attempted to be designed for the first time,respectively subject to impulsive time sequence and state triggering set.In order to save time and control cost,both problems of minimum-time control and minimum-triggering control are considered,and the time-optimal and triggering-optimal impulsive control strategies are given successively.The designed impulsive control method is applicable to several kinds of finite-field networks,including traditional switching ones,probabilistic one,and Markov jumping ones;and it can be applied to address stabilization and synchronization problems.In future work,how to design the event-triggered controller,sampled-data controller and impulsive controller with low computational complexity for large-scale Boolean networks is still the research focus.For finite-field networks,open issues remain in giving a unified framework for control design,hence it deserves further study.