Some Topics On The Control Theory Of Boolean Networks

Boolean networks are recently attracting considerable interest as important mod-els for biological systems and, in particular, as models of gene regulating networks. According to the actual background, many complex gene regulating networks can be described as Boolean networks in which each gene is regarded as a vertex of the net-work and interacted with each other. The recent decades have witnessed the great development of researches on the study of Boolean networks, which plays a significant role in exploring the mechanism of life activities and causes of disease. With the devel-opment of the systems biology, the analysis and control of Boolean networks become a hot topic for multidisciplinary research. Boolean network is a logical dynamic system, and it is difficult to deal with the large scale gene regulating network because of the lack of suitable tools. In this paper, with the help of semi-tensor product method, the dynamic equation of a Boolean network can be written in the form of a discrete-time linear system, which facilitates the analysis of Boolean networks. To be concrete, the contributions of this dissertation are as follows:In the real world, many evolutionary processes especially the biological systems may experience abrupt changes of states at certain time instants. These maybe due to changes in the interconnections of subsystems, sudden environment changes, etc. To describe mathematically an evolution of a real process with a short-term perturbation, it is sometimes convenient to consider these perturbations in the form of impulses. In biological systems, some states may correspond to unfavorable of dangerous situations. In chapter three, we first introduce two definitions of controllability:controllable at s-step and controllable avoiding forbidden states C. Then we study the controllability of Boolean networks with impulsive effects and forbidden states, and some necessary and sufficient conditions are obtained for the controllability.In chapter four, we investigate the optimal control and minimum-time control of a Boolean network with impulsive disturbances, where the impulses are time variant. Fix a vector r∈R2n, and consider the cost-functional J(u)=rTx(N; u), i.e. a Mayer- type optimal control problem, we provide a necessary condition for optimality in terms of the switching functions αi. Note that this is somewhat similar to the Pontryagin’s maximum principle for discrete-time dynamical systems. Then we investigate the minimum-time control of a Boolean network with impulsive disturbances, we derive several necessary conditions, stated in the form of maximum principles, for a control to be time-optimal. Finally, an biological example is provided to illustrate the efficiency of the obtained results.A probabilistic Boolean network is a randomized Boolean network, and it shares the appealing properties of Boolean network and also copes with the presence of un-certainty. We study the controllability of probabilistic Boolean network in chapter five. First, an algebraic expression of the probabilistic Boolean networks is obtained by the semi-tensor product of matrices. Then we give a simple algebraic formula for the transition probability between given initial and final states. Finally, we construct the controllability matrix based on a new operator, and some necessary and sufficient conditions are obtained for the controllability and reachability of probabilistic Boolean networks.In chapter six, we consider the solvability and controllability of singular Boolean networks, the solvability of singular Boolean networks is equivalent to the solvability of corresponding algebraic equation based on a new conversion. Then some necessary and sufficient conditions are given for the solvability of singular Boolean networks. Finally, controllability of singular Boolean control networks is studied under certain assumption, and several necessary and sufficient conditions are obtained as well.In chapter seven, we investigates the complete synchronization of two Boolean networks via logic control. One of the two networks is called the master network, while the other with control input is called the slave network, and the above two net-works have the same dimension. We first consider the synchronization of two Boolean networks via feedback control u(t)=Hx(t)y(t). Based on semi-tensor product of matrices, we derive the synchronization criteria expressed in an algebraic form. Par-ticularly, let u(t)=x(t), then the system we consider can be reduced to drive-response system. Hence, the system we consider is more general. Then we consider the complete synchronization of two Boolean networks via open loop control, and some necessary and sufficient conditions are obtained for the complete synchronization.In chapter eight, we study the stability of Boolean networks and the stabiliza-tion of Boolean control networks with respect to part of the system’s states X=(x1,..., xr). First, we give the definition of stability and stabilization of Boolean net-works. Then some necessary and sufficient conditions for partial stability of Boolean networks are given. Finally, the stabilization of Boolean control networks by a free control sequence and a state feedback control are investigated and the respective nec-essary and sufficient conditions are obtained. It is interesting to find that although the Boolean networks may be unstable (in the standard concept), it is partially stable. Examples are also provided to illustrate the efficiency of the obtained results.