Stability,Stabilization And Output Tracking Control Of Stochastic Switched Boolean Networks

Boolean networks were proposed by Kauffman in 1969 for modeling gene regulatory networks.Although Boolean network is a rough simplification of gene regulatory network and genetic reality,it can correctly capture the correct pattern of expression and suppression genes in some cases.Considering the randomness of gene expression and the noise in laboratory data measurements,Shmulevich et al.proposed a kind of stochastic switched Boolean networks called probabilistic Boolean networks in 2002.Boolean networks have been receiving widespread attention since their inception,and their dynamic behavior can be used to simulate many biologically significant phenomena.Thus they have been widely applied in many fields,such as cell differentiation,apoptosis,gene regulation,and so on.There were few uniform tools to study Boolean networks in the past.Since the semi-tensor product of matrices was put forward,this tool has greatly promoted the study of dynamic analysis and control problems of Boolean networks.Based on semi-tensor product,this thesis mainly studies the stability,stabilization,and output tracking problem of two kinds of stochastic switched Boolean networks:probabilistic Boolean networks and Markovian jumping Boolean networks.The main results include the following aspects:The global asymptotic stabilization of probabilistic Boolean networks under pinning control is investigated.Based on the state transition graph,a construction method of pinning controllers for the global asymptotic stabilization of probabilistic Boolean networks is obtained.Its essence is to change the state trajectories through pinning control and construct a directed spanning tree with the target vertex as the root node on the state transition graph.In addition,a comparison is made between the modeindependent pinning control and the mode-dependent pinning control,and it is proven that in order to globally asymptotically stabilize the system,the former requires no more pinning nodes and control inputs than the latter.Moreover,a necessary and sufficient condition is obtained to determine whether the global asymptotic stabilization of the probabilistic Boolean network can be achieved by controlling a given set of pinning nodes.Based on this necessary and sufficient condition,some algorithms are proposed to find the minimum set of pinning nodes that meets the condition.Compared with the existing results,the total computational complexity of these algorithms is reduced.Finally,the conclusions are extended to the case where the stabilization objective is a limit cycle.The global asymptotic set stability of probabilistic Boolean networks under random impulsive disturbances is investigated,where the impulsive probabilistic Boolean networks are described by a kind of hybrid index models.Sampling the instantaneous states of the impulsive probabilistic Boolean network at impulsive instants,the resulting state subsequence is a finite-state homogeneous Markov chain,and its convergence can be judged based on the principle of invariant sets.Based on the index of cyclicity and index of convergence of a Boolean matrix,an approximation calculation method of the transition probability matrix of this Markov chain is given,and the approximation result obtained does not affect the judgment of the convergence of the Markov chain.By analyzing the relationship between the asymptotic set stability of the impulsive probabilistic Boolean network and the asymptotic convergence of the Markov chain subsequence,the necessary and sufficient conditions for the global asymptotic set stability of the impulsive probabilistic Boolean network in the hybrid domain and time domain are obtained,respectively.Several kinds of output tracking problems of probabilistic Boolean control networks are investigated.For the asymptotic tracking control problem of probabilistic Boolean control networks,considering the cases where the tracking target is a constant signal and the output trajectory of a reference system,the asymptotic tracking control problem is transformed into the asymptotic stabilization problem of the system and the asymptotic stabilization problem of an augmented system,respectively.Then the sufficient and necessary criteria for the asymptotic tracking of the probabilistic Boolean control networks are obtained,and the design methods of corresponding state feedback controllers and output state feedback controllers are provided,respectively.Besides,aiming at the situation that the tracking target is a predetermined trajectory with finite length,an optimal control strategy is designed through dynamic programming algorithm to minimize the total average error of the probabilistic Boolean control network tracking the predetermined trajectory.In addition,considering the cost of changes in control inputs,a new objective function is obtained,and then an optimal control strategy is designed based on an auxiliary variable and dynamic programming algorithm to minimize the expected value of this objective function.The stability and stabilization of Markovian jumping Boolean networks are investigated.Based on the limit theory of finite-state homogeneous Markov chains,the existing conclusions on the asymptotic stability of Markovian jumping Boolean networks are supplemented,and the asymptotic stability criteria of two kinds of Markovian jumping Boolean networks with time delays are obtained.Moveover,based on the asymptotic convergence set and the finite-time convergence set of the switching signal sequence,the criteria for the asymptotic stabilization and the finite-time stabilization of Markovian jumping Boolean control networks under sampled-data control are obtained respectively,and the design methods of sampled-data controllers are provided.Furthermore,based on the state transition graph method,the asymptotic stabilization criterion of the Markovian jumping Boolean control network under a kind of event-triggered control is obtained.The Edmonds’ algorithm is used to obtain the minimum event-triggered set and the corresponding controller that asymptotically stabilizes the system.Finally,the event-triggered control method is extended to the case of systems with time delays.