Data based identification and prediction of nonlinear and complex dynamical systems

We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin

Pinning controllability of autonomous Boolean control networks

Autonomous Boolean networks (ABNs), which are developed to model the Boolean networks (BNs) with regulatory delays, are well known for their advantages of characterizing the intrinsic evolution rules of biological systems such as the gene regulatory networks. As a special type of ABNs with binary inputs, the autonomous Boolean control networks (ABCNs) are introduced for designing and analyzing the therapeutic intervention strategies where the binary inputs represent whether a certain medicine is dominated or not. An important problem in the therapeutic intervention is to design a control sequence steering an ABCN from an undesirable location (implying a diseased state) to a desirable one (corresponding to a healthy state). Motivated by such background, this paper aims to investigate the reachability and controllability of ABCNs with pinning controllers. Several necessary and sufficient criteria are provided by resorting to the semi-tensor product techniques of matrices. Moreover, an effective pinning control algorithm is presented for steering an ABCN from any given states to the desired state in the shortest time period. Numerical examples are also presented to demonstrate the results obtained.This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61329301, 61273156), Natural Science Foundation of Jiangsu Province of China (Grant No. BK20130017), Six Talent Peaks Project for the High Level Personnel from the Jiangsu Province of China (Grant No. 2015- DZXX-003), Scientific Research Foundations of Graduate School of Southeast University (Grant No. YBJJ1560), and Innovation Program of Jiangsu Province (Grant No. KYZZ15 0050)

Stability Analysis and Control of Several Classes of Logical Dynamic Systems and the Applications in Game Theory

With the rapid development of complex networks, logical dynamic systems have been commonly used mathematical models for simulating Genetic Regulatory Networks (GRNs) and Networked Evolutionary Games (NEGs), which have attracted considerable attention from biology, economy and many other fields. By resorting to the Semi-Tensor Product (STP) of matrices, logical dynamic systems can be equivalently converted into discrete time linear systems with algebraic forms. Based on that, this thesis analyzes the stability and studies the control design problems of several classes of logical dynamic systems. Moreover, the obtained results are applied to investigate the control and optimization problems of NEGs. The main results of this thesis are the following. • The stability and event-triggered control for a class of k-Valued Logical Networks (KVLNs) with time delays are studied. First, some necessary and sufficient con- ditions are obtained to detect the stability of Delayed k-Valued Logical Networks (DKVLNs). Second, the global stabilization problem under event-triggered control is considered, and some necessary and sufficient conditions are presented for the sta- bilization of Delayed k-Valued Logical Control Networks (DKVLCNs). Moreover, an algorithm is proposed to construct all the event-triggered state feedback controllers via antecedence solution technique. • The robust control invariance and robust set stabilization problems for a class of Mix- Valued Logical Control Networks (MVLCNs) with disturbances are studied. First, a calculation method for the Largest Robust Control Invariant Set (LRCIS) contained in a given set is introduced. Second, based on the Robust Control Invariant Subset (RCIS) obtained, the robust set stabilization of MVLCNs is discussed, and some new results are presented. Furthermore, the design algorithm of time-optimal state feedback stabilizers via antecedence solution technique is derived. • The robust set stability and robust set stabilization problems for a class of Probabilis- tic Boolean Control Networks (PBCNs) with disturbances are studied. An algorithm to determine the Largest Robust Invariant Set (LRIS) with probability 1 of a given set for a Probabilistic Boolean Network (PBN) is proposed, and the necessary and sufficient conditions to detect whether the PBN is globally finite-time stable to this invariant set with probability 1 are established. Then, the PBNs with control inputs are considered, and an algorithm for LRCIS with probability 1 is provided, based on which, some necessary and sufficient conditions for finite-time robust set stabiliza- tion with probability 1 of PBCNs are presented. Furthermore, the design scheme of time-optimal state feedback stabilizers via antecedence solution technique is derived. • The stabilization and set stabilization problems for a class of Switched Boolean Con- trol Networks (SBCNs) with periodic switching signal are studied. First, algebraic forms are constructed for SBCNs with periodic switching signal. Second, based on the algebraic formulations, the stabilization and set stabilization of SBCNs with peri- odic switching signal are discussed, and some new results are presented. Furthermore, constructive procedure of open loop controllers is given, and the design algorithms of switching-signal-dependent state feedback controllers via antecedence solution tech- nique are derived. • The dynamics and control problems for a class of NEGs with time-invariant delay in strategies are studied. First, algebraic forms are constructed for Delayed Networked Evolutionary Games (DNEGs). Second, based on the algebraic formulations, some necessary and sufficient conditions for the global convergence of desired strategy pro- file under a state feedback event-triggered controller are presented. Furthermore, the constructive procedure and the number of all valid event-triggered state feedback controllers are derived, which can make the game converge globally. • The evolutionary dynamics and optimization problems of the networked evolutionary boxed pig games with the mechanism of passive reward and punishment are studied. First, an algorithm is provided to construct the algebraic formulation for the dynamics of this kind of games. Then, the impact of reward and punishment parameters on the final cooperation level of the whole network is discussed

On the effects of firing memory in the dynamics of conjunctive networks

Boolean networks are one of the most studied discrete models in the context of the study of gene expression. In order to define the dynamics associated to a Boolean network, there are several \emph{update schemes} that range from parallel or \emph{synchronous} to \emph{asynchronous.} However, studying each possible dynamics defined by different update schemes might not be efficient. In this context, considering some type of temporal delay in the dynamics of Boolean networks emerges as an alternative approach. In this paper, we focus in studying the effect of a particular type of delay called \emph{firing memory} in the dynamics of Boolean networks. Particularly, we focus in symmetric (non-directed) conjunctive networks and we show that there exist examples that exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in determinate if some vertex will eventually change its state, given an initial condition. We prove that this problem is {\bf PSPACE}-complete

A weighted pair graph representation for reconstructibility of Boolean control networks

A new concept of weighted pair graphs (WPGs) is proposed to represent a new reconstructibility definition for Boolean control networks (BCNs), which is a generalization of the reconstructibility definition given in [Fornasini & Valcher, TAC2013, Def. 4]. Based on the WPG representation, an effective algorithm for determining the new reconstructibility notion for BCNs is designed with the help of the theories of finite automata and formal languages. We prove that a BCN is not reconstructible iff its WPG has a complete subgraph. Besides, we prove that a BCN is reconstructible in the sense of [Fornasini & Valcher, TAC2013, Def. 4] iff its WPG has no cycles, which is simpler to be checked than the condition in [Fornasini & Valcher, TAC2013, Thm. 4].Comment: 20 pages, 10 figures, accepted by SIAM Journal on Control and Optimizatio

Robust set stabilization of Boolean control networks with impulsive effects

This paper addresses the robust set stabilization problem of Boolean control networks (BCNs) with impulsive effects via the semi-tensor product method. Firstly, the closed-loop system consisting of a BCN with impulsive effects and a given state feedback control is converted into an algebraic form. Secondly, based on the algebraic form, some necessary and sufficient conditions are presented for the robust set stabilization of BCNs with impulsive effects under a given state feedback control and a free-form control sequence, respectively. Thirdly, as applications, some necessary and sufficient conditions are presented for robust partial stabilization and robust output tracking of BCNs with impulsive effects, respectively. The study of two illustrative examples shows that the obtained new results are effective