概率级联布尔网络的集镇定及其应用
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摘要
随着系统生物学和医学的迅速发展,基因调控网络已经成为一个热点研究领域.布尔网络作为研究生物系统和基因调控网络的一种重要模型,近年来引起了包括生物学家和系统科学家在内的很多学者的广泛关注.本文利用代数状态空间方法,研究了概率级联布尔网络的集镇定问题.首先给出概率级联布尔网络集镇定的定义,并利用矩阵的半张量积给出了概率级联布尔网络的代数表示.其次基于该代数表示,定义了一组合适的概率能达集,并给出了概率级联布尔网络集镇定问题可解的充要条件.最后将所得的理论结果应用于概率级联布尔网络的同步分析及n人随机级联演化布尔博弈的策略一致演化行为分析.
With the rapid development of systems biology and medical science, gene regulatory networks have become a heated research field. As an important model for studying biological systems and gene regulatory networks, in the past few decades, Boolean networks have attracted extensive attention from many scholars including biologists and system scientists.This paper studies the set stabilization problem of probabilistic cascading Boolean networks(PCBNs). Firstly, the concept of set stabilization of PCBNs has been proposed, and the considered PCBN is converted to an equivalent algebraic form by using the semi-tensor product of matrices. Secondly, based on the equivalent algebraic form, a series of probabilistic reachable sets is defined and a necessary and sufficient condition is presented for the set stabilization of PCBNs. Finally,as applications of set stabilization of PCBNs, the synchronization of PCBNs and strategy consensus of n-person random cascading evolutionary Boolean games are investigated, respectively, and several necessary and sufficient conditions are presented.
With the rapid development of systems biology and medical science, gene regulatory networks have become a heated research field. As an important model for studying biological systems and gene regulatory networks, in the past few decades, Boolean networks have attracted extensive attention from many scholars including biologists and system scientists.This paper studies the set stabilization problem of probabilistic cascading Boolean networks(PCBNs). Firstly, the concept of set stabilization of PCBNs has been proposed, and the considered PCBN is converted to an equivalent algebraic form by using the semi-tensor product of matrices. Secondly, based on the equivalent algebraic form, a series of probabilistic reachable sets is defined and a necessary and sufficient condition is presented for the set stabilization of PCBNs. Finally,as applications of set stabilization of PCBNs, the synchronization of PCBNs and strategy consensus of n-person random cascading evolutionary Boolean games are investigated, respectively, and several necessary and sufficient conditions are presented.
引文
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[14] LI H T, WANG Y Z, XIE L H. Output tracking control of Boolean control networks via state feedback:Constant reference signal case.Automatica, 2015, 59:54–59.
[15] LI H T, XIE L H, WANG Y Z. On robust control invariance of Boolean control networks. Automatica, 2016, 68:392–396.
[16] LI R, YANG M, CHU T G. State feedback stabilization for Boolean control networks. IEEE Transactions Automatic Control, 2013, 58(7):1853–1857.
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[19] MENG M, FENG J E. Topological structure and the disturbance decoupling problem of singular Boolean networks. IET Control Theory&Applications, 2014, 8(13):1247–1255.
[20] ZHANG L J, ZHANG K Z. Controllability and observability of Boolean control networks with time-variant delays in states. IEEE Transactions on Neural Networks and Learning Systems, 2013, 24(9):1478–1484.
[21] ZHONG J, LU J Q, LIU Y, et al. Synchronization in an array of output-coupled Boolean networks with time delay. IEEE Transactions on Neural Networks and Learning Systems, 2014, 25(12):2288–2294.
[22] ZOU Y L, ZHU J D. Kalman decomposition for Boolean control networks. Automatica, 2015, 54:65–71.
[23] LIU Y, CHEN H W, LU J Q, et al. Controllability of probabilistic Boolean control networks based on transition probability matrices.Automatica, 2015, 52:340–345.
[24] WU Y H, SHEN T L. An algebraic expression of finite horizon optimal control algorithm for stochastic logical dynamical systems. Systems&Control Letters, 2015, 82(3):108–114.
[25] HAN Ji, ZHANG Huaguang, TIAN Hui. Complete synchronization of the periodically time-variant Boolean networks. Control Theory&Applications, 2016, 33(7):863–869.(韩吉,张化光,田辉.周期时变布尔网络的完全同步化.控制理论与应用, 2016, 33(7):863–869.)
[26] LI H T, ZHAO G D, MENG M, et al. A survey on applications of semi-tensor product method in engineering. Science China Information Sciences, 2018, 61(1):010202.
[27] LI R, YANG M, CHU T G. State feedback stabilization for probabilistic Boolean network. Automatica, 2014, 50(4):1272–1278.
[28] LU J Q, LI H T, LIU Y, et al. Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems. IET Control Theory&Applications, 2017, 11(13):2040–2047.
[29] SONG Pingping, LI Haitao, YANG Qiqi, et al. Pinning output tracking control of multi-valued logical networks. Control Theory&Applications, 2016, 33(9):1200–1206.(宋平平,李海涛,杨琪琪,等.多值逻辑网络的输出跟踪牵制控制.控制理论与应用, 2016, 33(9):1200–1206.)
[30] ARACENA J, DEMONGEOT J, FANCHON E. On the number of different dynamics in Boolean networks with deterministic update schedules. Mathematical Biosciences, 2013, 242(2):188–194.
[31] ALBERT R, ROBEVA R. Algebraic and Discrete Mathematical Methods for Modern Biology. Elsevier:Amsterdam, 2015.
[32] CHENG D Z, HE F, QI H S. Modeling, analysis and control of networked evolutionary games. IEEE Transactions on Automatic Control, 2015, 60(9):2402–2415.
[33] LUO C, WANG X Y, LIU H. Controllability of time-delayed Boolean multiplex control networks under asynchronous stochastic update.Scientific Reports, 2014, 4(1):7522.
[34] ROBERT F. Discrete Iterations:A Metric Study, Series in Computational Mathematics. Berlin:Springer, 1986.
[35] TOURNIER L, CHAVES M. Uncovering operational interactions in genetic networks using asynchronous Boolean dynamics. Journal of Theoretical Biology, 2009, 260(2):196–209.
[36] YANG M, CHU T G. Evolution of attractors and basins of asynchronous random Boolean networks. Physical Review E, 2012, 85(2):976–986.
[37] ZHANG H, WANG X Y, LIN X H. Synchronization of asynchronous switched Boolean network. IEEE/ACM Transactions on Computational Biology&Bioinformatics, 2015, 12(6):1449–1456.
[38] YANG Q Q, LI H T, SONG P P, et al. Global convergence of serial Boolean networks based on algebraic representation. Journal of Difference Equations and Applications, 2016, 23(3):1–15.
[2] SHMULEVICH I, DOUGHERTY E, KIM S, et al. Probabilistic Boolean networks:a rule-based uncertainty model for gene regulatory networks. Bioinformatics, 2002, 18(2):261–274.
[3] DING X Y, LI H T, YANG Q Q, et al. Stochastic stability and stabilization of n-person random evolutionary Boolean games. Applied Mathematics and Computation, 2017, 306:1–12.
[4] HAYASHIDA M, TAMURA T, AKUTSU T, et al. Disturbance and enumeration of attractors in probabilistic Boolean networks. IET Systems Biology, 2009, 3(6):465–474.
[5] PAL R, DATTA A, DOUGHERTY E. Optimal infinite horizon control for probabilistic Boolean networks. IEEE Transactions Signal Process, 2006, 54(6):2375–2387.
[6] TRAIRATPHISAN P, MIZERA A, PANG J, et al. Recent development and biomedical applications of probabilistic Boolean networks.Cell Communication Signal, 2013, 11(46):1–25.
[7] CHENG D Z, QI H S, LI Z Q. Analysis and Control of Boolean Networks:A Semi-tensor Product Approach. Springer:London, 2011.
[8] CHENG D Z, QI H S, ZHAO Y. An Introduction to Semi-tensor Product of Matrices and Its Applications. Singapore:World Scientific,2012.
[9] CHEN H, SUN J. Output controllability and optimal output control of state-dependent switched Boolean control networks. Automatica,2014, 50(7):1929–1934.
[10] FORNASINI E, VALCHER M. Observability, reconstructibility and state observers of Boolean control networks. IEEE Transactions on Automatic Control, 2013, 58(6):1390–1401.
[11] LASCHOV D, MARGALIOT M. Controllability of Boolean control networks via the Perron-Frobenius theory. Automatica, 2012, 48(6):1218–1223.
[12] LI F F, TANG Y. Set stabilization for switched Boolean control networks. Automatica, 2017, 78:223–230.
[13] LI Haitao, WANG Yuzhen. Stability analysis for switched singular Boolean networks. Control Theory&Applications, 2014, 31(7):908–914.(李海涛,王玉振.切换奇异布尔网络的稳定性分析.控制理论与应用, 2014, 31(7):908–914.)
[14] LI H T, WANG Y Z, XIE L H. Output tracking control of Boolean control networks via state feedback:Constant reference signal case.Automatica, 2015, 59:54–59.
[15] LI H T, XIE L H, WANG Y Z. On robust control invariance of Boolean control networks. Automatica, 2016, 68:392–396.
[16] LI R, YANG M, CHU T G. State feedback stabilization for Boolean control networks. IEEE Transactions Automatic Control, 2013, 58(7):1853–1857.
[17] LIU Y, CHEN H W, WU B. Controllability of Boolean control networks with impulsive effects and forbidden states. Mathematical Methods in the Applied Sciences, 2014, 31(1):1–9.
[18] LU J Q, LI M L, LIU Y, et al. Nonsingularity of Grain-like cascade FSRs via semi-tensor product. Science China Information Sciences,2018, 61(1):010204.
[19] MENG M, FENG J E. Topological structure and the disturbance decoupling problem of singular Boolean networks. IET Control Theory&Applications, 2014, 8(13):1247–1255.
[20] ZHANG L J, ZHANG K Z. Controllability and observability of Boolean control networks with time-variant delays in states. IEEE Transactions on Neural Networks and Learning Systems, 2013, 24(9):1478–1484.
[21] ZHONG J, LU J Q, LIU Y, et al. Synchronization in an array of output-coupled Boolean networks with time delay. IEEE Transactions on Neural Networks and Learning Systems, 2014, 25(12):2288–2294.
[22] ZOU Y L, ZHU J D. Kalman decomposition for Boolean control networks. Automatica, 2015, 54:65–71.
[23] LIU Y, CHEN H W, LU J Q, et al. Controllability of probabilistic Boolean control networks based on transition probability matrices.Automatica, 2015, 52:340–345.
[24] WU Y H, SHEN T L. An algebraic expression of finite horizon optimal control algorithm for stochastic logical dynamical systems. Systems&Control Letters, 2015, 82(3):108–114.
[25] HAN Ji, ZHANG Huaguang, TIAN Hui. Complete synchronization of the periodically time-variant Boolean networks. Control Theory&Applications, 2016, 33(7):863–869.(韩吉,张化光,田辉.周期时变布尔网络的完全同步化.控制理论与应用, 2016, 33(7):863–869.)
[26] LI H T, ZHAO G D, MENG M, et al. A survey on applications of semi-tensor product method in engineering. Science China Information Sciences, 2018, 61(1):010202.
[27] LI R, YANG M, CHU T G. State feedback stabilization for probabilistic Boolean network. Automatica, 2014, 50(4):1272–1278.
[28] LU J Q, LI H T, LIU Y, et al. Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems. IET Control Theory&Applications, 2017, 11(13):2040–2047.
[29] SONG Pingping, LI Haitao, YANG Qiqi, et al. Pinning output tracking control of multi-valued logical networks. Control Theory&Applications, 2016, 33(9):1200–1206.(宋平平,李海涛,杨琪琪,等.多值逻辑网络的输出跟踪牵制控制.控制理论与应用, 2016, 33(9):1200–1206.)
[30] ARACENA J, DEMONGEOT J, FANCHON E. On the number of different dynamics in Boolean networks with deterministic update schedules. Mathematical Biosciences, 2013, 242(2):188–194.
[31] ALBERT R, ROBEVA R. Algebraic and Discrete Mathematical Methods for Modern Biology. Elsevier:Amsterdam, 2015.
[32] CHENG D Z, HE F, QI H S. Modeling, analysis and control of networked evolutionary games. IEEE Transactions on Automatic Control, 2015, 60(9):2402–2415.
[33] LUO C, WANG X Y, LIU H. Controllability of time-delayed Boolean multiplex control networks under asynchronous stochastic update.Scientific Reports, 2014, 4(1):7522.
[34] ROBERT F. Discrete Iterations:A Metric Study, Series in Computational Mathematics. Berlin:Springer, 1986.
[35] TOURNIER L, CHAVES M. Uncovering operational interactions in genetic networks using asynchronous Boolean dynamics. Journal of Theoretical Biology, 2009, 260(2):196–209.
[36] YANG M, CHU T G. Evolution of attractors and basins of asynchronous random Boolean networks. Physical Review E, 2012, 85(2):976–986.
[37] ZHANG H, WANG X Y, LIN X H. Synchronization of asynchronous switched Boolean network. IEEE/ACM Transactions on Computational Biology&Bioinformatics, 2015, 12(6):1449–1456.
[38] YANG Q Q, LI H T, SONG P P, et al. Global convergence of serial Boolean networks based on algebraic representation. Journal of Difference Equations and Applications, 2016, 23(3):1–15.