Theory And Applications Of Logical Networks Via Semi-tensor Product

Based on the semi-tensor product of matrices, logical networks can be equivalently converted into an algebraic form. In this dissertation, logical networks and singular Boolean networks are investigated via semi-tensor product. Synchronization of multi-valued logical networks, function pertur-bations and l1-gain and model reduction problems for Boolean networks are discussed. Topological structure for singular Boolean networks is proposed, and the control problems for singular Boolean networks, including control-lability, observability, optimal control and disturbance decoupling problems, are studied. There are eight chapters in this dissertation.In Chapter 1, the research background and current status for logical networks and singular Boolean networks are introduced. The definition and properties of semi-tensor product are given in Chapter 2, including the alge-braic expressions of logical functions and logical networks.In Chapter 3, synchronization problems for interconnected logical net-works and higher logical networks are analyzed, respectively. By giving the algebraic expression of interconnected logical networks via semi-tensor product, a necessary and sufficient condition for synchronization of k-valued logical networks is obtained according to the definition of synchronization. Combining with the characteristics of fixed points and limit cycles of logical networks, another necessary and sufficient condition for synchronization is constructed. Then, a third equivalent condition of synchronization is derived by analyzing the relation between limit set and transition matrix. Final- ly, the obtained results are generalized to higher order multi-valued logical networks.In Chapter 4, the impacts of function perturbations in Boolean networks on fixed points and limit cycles are investigated. Two function perturbation-s, one-bit perturbation and modifications of update schedule, are discussed. The algebraic form of perturbed Boolean networks under one-bit perturba-tion is given, based on which several necessary and sufficient conditions of different kinds of effects on state transition and attractors are obtained. Be-sides, a Boolean network with update schedule is studied and the definition of its adjacent diagraph is first presented in this chapter. Furthermore, tran-sition matrix of the updated network is computed to analyze the changes of fixed points and limit cycles. Finally, identifying function perturbations with its application to Drosophila melanogaster gene networks is shown to demonstrate the practicability and effectiveness of the theoretical results.In Chapter 5,li-gain analysis and h model reduction problem are pro-posed and discussed. First, the input energy and output energy are described as pseudo-Boolean functions, based on which the definition of weighted l1-gain is introduced. By constructing a co-positive Lyapunov function, a suf-ficient condition is established to ensure that a Boolean network is not only internally asymptotically stable, but also has an l1-gain no more than a given scalar. Along this line, the l1 model reduction problem is presented and con-verted to the l1-gain problem of another Boolean control network with more nodes. At the end of this chapter, two illustrative examples are displayed to show the feasibility of the theoretical results.In Chapter 6, the general singular Boolean networks are proposed, and the solvability, fixed points and limit cycles problems are discussed. Based on the algebraic form of singular Boolean networks, the admissible condition set is introduced, and a necessary and sufficient condition of unique existence of solution of singular Boolean networks is obtained. In order to calculate the fixed points and limit cycles, the generalized transition matrix of a singular Boolean network is defined, which contains all the state transferring informa-tion. Similar to the results of Boolean networks, some equivalent conditions for fixed points and limit cycles are derived. Finally, illustrative examples are given to show the feasibility of the results.In Chapter 7, the control problems for singular Boolean control net-works are studied, such as controllability, observability, optimal control and disturbance decoupling problems. The input-state incidence matrix of sin-gular Boolean control networks is presented, based on which the generalized input-state incidence matrix is also defined. Similar to Boolean control net-works, a necessary and sufficient condition for the controllability of singular Boolean control networks and a sufficient condition for observability of singu-lar Boolean control networks are given. Using an analogous needle variation, for multi-input case, a necessary condition for the existence of optimal con-trol is provided, and the result is specialized to the single-input case. Finally, the disturbance decoupling problem for singular Boolean control networks is presented and solved by a constant control.Chapter 8 concludes this dissertation and lists some prospects of study about logical networks.