State feedback for set stabilization of Markovian jump Boolean control networks

Sanmei Zhu; Jun-e Feng; Jianli Zhao

doi:10.3934/dcdss.2020413

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doi: 10.3934/dcdss.2020413

State feedback for set stabilization of Markovian jump Boolean control networks

Sanmei Zhu ^1, , Jun-e Feng ^1,, and Jianli Zhao ^2,

1.	School of Mathematics, Shandong University, Jinan, China
2.	STP Center, Liaocheng University, Liaocheng, China

^* Corresponding author: Jun-e Feng

Received January 2020 Revised April 2020 Published July 2020

In this paper, the set stabilization problem of Markovian jump Boolean control networks (MJBCNs) is investigated via semi-tensor product of matrices. First, the conception of set stabilization is proposed for MJBCNs. Then based on the algebraic expression of MJBCN, a necessary and sufficient condition for set stabilization is provided by a linear programming problem, which is simple to solve. Moreover, by solving this linear programming problem, an algorithm for designing a state feedback controller is developed. Finally, two examples are presented to illustrate the feasibility of the obtained results.

Keywords: Set stabilization, Markovian jump Boolean control networks, semi-tensor product, state feedback, control invariant subset.

Mathematics Subject Classification: Primary: 93D15, 93E03.

Citation: Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020413

References:

[1]	H. Bei, L. Wang, Y. Ma, J. Sun and L. Zhang, A linear optimal feedback control for producing $1, 3$-propanediol via microbial fermentation, Discrete and Continous Dynamical Systems series S, 13 (2020), 1623-1635. doi: 10.3934/dcdss.2020095. Google Scholar
[2]	D. Cheng, H. Qi and Z. Li, Analysis and Control of Boolean networks: A Semi-Tensor Product Approach, Springer, 2011. doi: 10.1007/978-0-85729-097-7. Google Scholar
[3]	D. Cheng and H. Qi, Controllability and observability of Boolean control networks, Automatica, 45 (2009), 1659-1667. doi: 10.1016/j.automatica.2009.03.006. Google Scholar
[4]	A. N. Churilov, Orbital stability of periodic solutions of an impulsive system with a linear continuous-time part, AIMS Mathematics, 5 (2019), 96-110. doi: 10.3934/math.2020007. Google Scholar
[5]	O. L. Costa, M. D. Fragoso and R. P. Marques, Discrete-time Markov Jump Linear Systems, Springer, 2005. doi: 10.1007/b138575. Google Scholar
[6]	E. Fornasini and M. E. Valcher, Optimal control of Boolean control networks, IEEE Transactions on Automatic Control, 59 (2014), 1258-1270. doi: 10.1109/TAC.2013.2294821. Google Scholar
[7]	D. G. Green, T. G. Leishman and S. Sadedin, The emergence of social consensus in Boolean networks, in 2007 IEEE Symposium on Artificial Life, (2007), 402–408. Google Scholar
[8]	Y. Guo, P. Wang, W. Gui and C. Yang, Set stability and set stabilization of Boolean control networks based on invariant subsets, Automatica, 61 (2015), 106-112. doi: 10.1016/j.automatica.2015.08.006. Google Scholar
[9]	Y. Guo, R. Zhou, Y. Wu, W. Gui and C. Yang, Stability and set stability in distribution of probabilistic Boolean networks, IEEE Transactions on Automatic Control, 64 (2019), 736-742. doi: 10.1109/TAC.2018.2833170. Google Scholar
[10]	X. Hu, C. Huang, J. Lu and J. Cao, Stabilization of boolean control networks with stochastic impulses, Journal of the Franklin Institute, 356 (2019), 7164-7182. doi: 10.1016/j.jfranklin.2019.06.039. Google Scholar
[11]	S. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22 (1969), 437-467. doi: 10.1016/0022-5193(69)90015-0. Google Scholar
[12]	R. K. Layek, A. Datta and E. R. Dougherty, From biological pathways to regulatory networks, Molecular Biosystems, 7 (2011), 843-851. doi: 10.1039/c0mb00263a. Google Scholar
[13]	F. Li and L. Xie, Set stabilization of probabilistic Boolean networks using pinning control, IEEE Transactions on Neural Networks and Learning Systems, 30 (2019), 2555-2561. doi: 10.1109/TNNLS.2018.2881279. Google Scholar
[14]	X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Applied Mathematics and Computation, 329 (2018), 14-22. doi: 10.1016/j.amc.2018.01.036. Google Scholar
[15]	X. Li, Daniel W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361-368. doi: 10.1016/j.automatica.2018.10.024. Google Scholar
[16]	F. Li and Y. Tang, Set stabilization for switched Boolean control networks, Automatica, 78 (2017), 223-230. doi: 10.1016/j.automatica.2016.12.007. Google Scholar
[17]	Y. Li, H. Li and W. Sun, Event-triggered control for robust set stabilization of logical control networks, Automatica, 95 (2018), 556-560. doi: 10.1016/j.automatica.2018.06.030. Google Scholar
[18]	P. Li, J. Lam and Z. Shu, ${H}_{\infty}$ positive filtering for positive linear discrete-time systems: An augmentation approach, IEEE Transactions on Automatic Control, 55 (2010), 2337-2342. doi: 10.1109/TAC.2010.2053471. Google Scholar
[19]	H. Li, Y. Wang and Z. Liu, Simultaneous stabilization for a set of Boolean control networks, Systems and Control Letters, 62 (2013), 1168-1174. doi: 10.1016/j.sysconle.2013.09.008. Google Scholar
[20]	F. Li and Z. Yu, Feedback control and output feedback control for the stabilisation of switched Boolean networks, International Journal of Control, 89 (2016), 337-342. doi: 10.1080/00207179.2015.1076938. Google Scholar
[21]	R. Liu, J. Lu, W. X. Zheng and J. Kurths, Output feedback control for set stabilization of Boolean control networks, IEEE Transactions on Neural Networks and Learning Systems, 31 (2020), 2129-2139. doi: 10.1109/TNNLS.2019.2928028. Google Scholar
[22]	X. Liu and J. Zhu, On potential equations of finite games, Automatica, 68 (2016), 245-253. doi: 10.1016/j.automatica.2016.01.074. Google Scholar
[23]	Y. Liu, H. Chen, J. Lu and B. Wu, Controllability of probabilistic Boolean control networks based on transition probability matrices, Automatica, 52 (2015), 340-345. doi: 10.1016/j.automatica.2014.12.018. Google Scholar
[24]	H. Lyu, W. Wang, X. Liu and Z. Wang, Modeling of multivariable fuzzy systems by semitensor product, IEEE Transactions on Fuzzy Systems, 28 (2020), 228-235. doi: 10.1109/TFUZZ.2019.2902820. Google Scholar
[25]	H. Lyu, W. Wang and X. Liu, Universal approximation of multi-variable fuzzy systems by semi-tensor product, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1109/TFUZZ.2019.2946512. Google Scholar
[26]	J. Lu, H. Li, Y. Liu and F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory and Applications, 11 (2017), 2040-2047. doi: 10.1049/iet-cta.2016.1659. Google Scholar
[27]	M. Meng, L. Liu and G. Feng, Stability and $l_1$ gain analysis of boolean networks with markovian jump parameters, IEEE Transactions on Automatic Control, 62 (2017), 4222-4228. doi: 10.1109/TAC.2017.2679903. Google Scholar
[28]	M. Meng, J. Lam, J. Feng and K. C. Cheung, Stability and stabilization of Boolean networks with stochastic delays, IEEE Transactions on Automatic Control, 64 (2019), 790-796. doi: 10.1109/TAC.2018.2835366. Google Scholar
[29]	M. Meng, G. Xiao, C. Zhai and G. Li, Controllability of Markovian jump Boolean control networks, Automatica, 106 (2019), 70-76. doi: 10.1016/j.automatica.2019.04.028. Google Scholar
[30]	N. Megiddo, Linear programming in linear time when the dimension is fixed, Journal of the ACM, 31 (1984), 114-127. doi: 10.1145/2422.322418. Google Scholar
[31]	K. Ratnavelu, M. Kalpana and P. Balasubramaniam, Asymptotic stability of Markovian switching genetic regulatory networks with leakage and mode-dependent time delays, Journal of the Franklin Institute, 353 (2016), 1615-1638. doi: 10.1016/j.jfranklin.2016.01.015. Google Scholar
[32]	R. Rakkiyappan, A. Chandrasekar, F. A. Rihan and S. Lakshmanan, Exponentialstate estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays, Mathematical Biosciences, 251 (2014), 30-53. doi: 10.1016/j.mbs.2014.02.008. Google Scholar
[33]	I. Shmulevich, E. R. Dougherty and W. Zhang, From boolean to probabilistic Boolean networks as models of genetic regulatory networks, Proceedings of the IEEE, 90 (2002), 1778-1792. doi: 10.1109/JPROC.2002.804686. Google Scholar
[34]	X. Tan, J. Cao, X. Li and A. Alsaedi, Leader-following mean squareconsensus of stochastic multi-agent systems with input delay via event-triggered control, IET Control Theory Applications, 12 (2017), 299-309. doi: 10.1049/iet-cta.2017.0462. Google Scholar
[35]	H. Tian and Y. Hou, State feedback design for set stabilization of probabilistic Boolean control networks, Journal of the Franklin Institute, 356 (2019), 4358-4377. doi: 10.1016/j.jfranklin.2018.12.027. Google Scholar
[36]	H. Tian, H. Zhang, Z. Wang and Y. Hou, Stabilization of k-valued logical control networks by open-loop control via the reverse-transfer method, Automatica, 83 (2017), 387-390. doi: 10.1016/j.automatica.2016.12.040. Google Scholar
[37]	B. Wang, J. Feng, H. Li and Y. Yu, On detectability of Boolean control networks, Nonlinear Analysis: Hybrid Systems, 36 (2020), 100859, 18 pp. doi: 10.1016/j.nahs.2020.100859. Google Scholar
[38]	S. Wang, J. Feng, Y. Yu and J. Zhao, Further results on dynamic-algebraic Boolean control networks, Science China Information Sciences, 62 (2019), 12208, 14 pp. doi: 10.1007/s11432-018-9447-4. Google Scholar
[39]	M. Xu, Y. Liu, J. Lou, Z. Wu and J. Zhong, Set stabilization of probabilistic Boolean control networks: A sampled-data control approach, IEEE Transactions on Cybernetics, 2019, 1–8. doi: 10.1109/TCYB.2019.2940654. Google Scholar
[40]	D. Yang, X. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305. doi: 10.1016/j.nahs.2019.01.006. Google Scholar
[41]	X. Yang, X. Li, Q. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences and Engineering, 15 (2018), 1495-1515. doi: 10.3934/mbe.2018069. Google Scholar
[42]	Y. Yu, J. Feng, J. Pan and D. Cheng, Block decoupling of Boolean control networks, IEEE Transactions on Automatic Control, 64 (2019), 3129-3140. doi: 10.1109/TAC.2018.2880411. Google Scholar
[43]	Y. Yu, M. Meng, and J. Feng, Observability of Boolean networks via matrix equations, Automatica, 111 (2020), 108621, 5 pp. doi: 10.1016/j.automatica.2019.108621. Google Scholar
[44]	Q. Zhang, J. Feng, J. Pan and J. Xia, Set controllability for switched Boolean control networks, Neurocomputing, 359 (2019), 476-482. doi: 10.1016/j.neucom.2019.05.087. Google Scholar
[45]	R. Zhou, Y. Guo and W. Gui, Set reachability and observability of probabilistic Boolean networks, Automatica, 106 (2019), 230-241. doi: 10.1016/j.automatica.2019.05.021. Google Scholar
[46]	S. Zhu, J. Lu, Y. Liu, T. Huang and J. Kurths, Output tracking of probabilistic Boolean networks by output feedback control, Information Sciences, 483 (2019), 96-105. doi: 10.1016/j.ins.2018.12.087. Google Scholar
[47]	R. Zhou and Y. Guo, Set stabilization in distribution of probabilistic Boolean control networks, in 2018 13th World Congress on Intelligent Control and Automation (WCICA), (2018), 274–279. Google Scholar

show all references

References:

[1]	H. Bei, L. Wang, Y. Ma, J. Sun and L. Zhang, A linear optimal feedback control for producing $1, 3$-propanediol via microbial fermentation, Discrete and Continous Dynamical Systems series S, 13 (2020), 1623-1635. doi: 10.3934/dcdss.2020095. Google Scholar
[2]	D. Cheng, H. Qi and Z. Li, Analysis and Control of Boolean networks: A Semi-Tensor Product Approach, Springer, 2011. doi: 10.1007/978-0-85729-097-7. Google Scholar
[3]	D. Cheng and H. Qi, Controllability and observability of Boolean control networks, Automatica, 45 (2009), 1659-1667. doi: 10.1016/j.automatica.2009.03.006. Google Scholar
[4]	A. N. Churilov, Orbital stability of periodic solutions of an impulsive system with a linear continuous-time part, AIMS Mathematics, 5 (2019), 96-110. doi: 10.3934/math.2020007. Google Scholar
[5]	O. L. Costa, M. D. Fragoso and R. P. Marques, Discrete-time Markov Jump Linear Systems, Springer, 2005. doi: 10.1007/b138575. Google Scholar
[6]	E. Fornasini and M. E. Valcher, Optimal control of Boolean control networks, IEEE Transactions on Automatic Control, 59 (2014), 1258-1270. doi: 10.1109/TAC.2013.2294821. Google Scholar
[7]	D. G. Green, T. G. Leishman and S. Sadedin, The emergence of social consensus in Boolean networks, in 2007 IEEE Symposium on Artificial Life, (2007), 402–408. Google Scholar
[8]	Y. Guo, P. Wang, W. Gui and C. Yang, Set stability and set stabilization of Boolean control networks based on invariant subsets, Automatica, 61 (2015), 106-112. doi: 10.1016/j.automatica.2015.08.006. Google Scholar
[9]	Y. Guo, R. Zhou, Y. Wu, W. Gui and C. Yang, Stability and set stability in distribution of probabilistic Boolean networks, IEEE Transactions on Automatic Control, 64 (2019), 736-742. doi: 10.1109/TAC.2018.2833170. Google Scholar
[10]	X. Hu, C. Huang, J. Lu and J. Cao, Stabilization of boolean control networks with stochastic impulses, Journal of the Franklin Institute, 356 (2019), 7164-7182. doi: 10.1016/j.jfranklin.2019.06.039. Google Scholar
[11]	S. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22 (1969), 437-467. doi: 10.1016/0022-5193(69)90015-0. Google Scholar
[12]	R. K. Layek, A. Datta and E. R. Dougherty, From biological pathways to regulatory networks, Molecular Biosystems, 7 (2011), 843-851. doi: 10.1039/c0mb00263a. Google Scholar
[13]	F. Li and L. Xie, Set stabilization of probabilistic Boolean networks using pinning control, IEEE Transactions on Neural Networks and Learning Systems, 30 (2019), 2555-2561. doi: 10.1109/TNNLS.2018.2881279. Google Scholar
[14]	X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Applied Mathematics and Computation, 329 (2018), 14-22. doi: 10.1016/j.amc.2018.01.036. Google Scholar
[15]	X. Li, Daniel W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361-368. doi: 10.1016/j.automatica.2018.10.024. Google Scholar
[16]	F. Li and Y. Tang, Set stabilization for switched Boolean control networks, Automatica, 78 (2017), 223-230. doi: 10.1016/j.automatica.2016.12.007. Google Scholar
[17]	Y. Li, H. Li and W. Sun, Event-triggered control for robust set stabilization of logical control networks, Automatica, 95 (2018), 556-560. doi: 10.1016/j.automatica.2018.06.030. Google Scholar
[18]	P. Li, J. Lam and Z. Shu, ${H}_{\infty}$ positive filtering for positive linear discrete-time systems: An augmentation approach, IEEE Transactions on Automatic Control, 55 (2010), 2337-2342. doi: 10.1109/TAC.2010.2053471. Google Scholar
[19]	H. Li, Y. Wang and Z. Liu, Simultaneous stabilization for a set of Boolean control networks, Systems and Control Letters, 62 (2013), 1168-1174. doi: 10.1016/j.sysconle.2013.09.008. Google Scholar
[20]	F. Li and Z. Yu, Feedback control and output feedback control for the stabilisation of switched Boolean networks, International Journal of Control, 89 (2016), 337-342. doi: 10.1080/00207179.2015.1076938. Google Scholar
[21]	R. Liu, J. Lu, W. X. Zheng and J. Kurths, Output feedback control for set stabilization of Boolean control networks, IEEE Transactions on Neural Networks and Learning Systems, 31 (2020), 2129-2139. doi: 10.1109/TNNLS.2019.2928028. Google Scholar
[22]	X. Liu and J. Zhu, On potential equations of finite games, Automatica, 68 (2016), 245-253. doi: 10.1016/j.automatica.2016.01.074. Google Scholar
[23]	Y. Liu, H. Chen, J. Lu and B. Wu, Controllability of probabilistic Boolean control networks based on transition probability matrices, Automatica, 52 (2015), 340-345. doi: 10.1016/j.automatica.2014.12.018. Google Scholar
[24]	H. Lyu, W. Wang, X. Liu and Z. Wang, Modeling of multivariable fuzzy systems by semitensor product, IEEE Transactions on Fuzzy Systems, 28 (2020), 228-235. doi: 10.1109/TFUZZ.2019.2902820. Google Scholar
[25]	H. Lyu, W. Wang and X. Liu, Universal approximation of multi-variable fuzzy systems by semi-tensor product, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1109/TFUZZ.2019.2946512. Google Scholar
[26]	J. Lu, H. Li, Y. Liu and F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory and Applications, 11 (2017), 2040-2047. doi: 10.1049/iet-cta.2016.1659. Google Scholar
[27]	M. Meng, L. Liu and G. Feng, Stability and $l_1$ gain analysis of boolean networks with markovian jump parameters, IEEE Transactions on Automatic Control, 62 (2017), 4222-4228. doi: 10.1109/TAC.2017.2679903. Google Scholar
[28]	M. Meng, J. Lam, J. Feng and K. C. Cheung, Stability and stabilization of Boolean networks with stochastic delays, IEEE Transactions on Automatic Control, 64 (2019), 790-796. doi: 10.1109/TAC.2018.2835366. Google Scholar
[29]	M. Meng, G. Xiao, C. Zhai and G. Li, Controllability of Markovian jump Boolean control networks, Automatica, 106 (2019), 70-76. doi: 10.1016/j.automatica.2019.04.028. Google Scholar
[30]	N. Megiddo, Linear programming in linear time when the dimension is fixed, Journal of the ACM, 31 (1984), 114-127. doi: 10.1145/2422.322418. Google Scholar
[31]	K. Ratnavelu, M. Kalpana and P. Balasubramaniam, Asymptotic stability of Markovian switching genetic regulatory networks with leakage and mode-dependent time delays, Journal of the Franklin Institute, 353 (2016), 1615-1638. doi: 10.1016/j.jfranklin.2016.01.015. Google Scholar
[32]	R. Rakkiyappan, A. Chandrasekar, F. A. Rihan and S. Lakshmanan, Exponentialstate estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays, Mathematical Biosciences, 251 (2014), 30-53. doi: 10.1016/j.mbs.2014.02.008. Google Scholar
[33]	I. Shmulevich, E. R. Dougherty and W. Zhang, From boolean to probabilistic Boolean networks as models of genetic regulatory networks, Proceedings of the IEEE, 90 (2002), 1778-1792. doi: 10.1109/JPROC.2002.804686. Google Scholar
[34]	X. Tan, J. Cao, X. Li and A. Alsaedi, Leader-following mean squareconsensus of stochastic multi-agent systems with input delay via event-triggered control, IET Control Theory Applications, 12 (2017), 299-309. doi: 10.1049/iet-cta.2017.0462. Google Scholar
[35]	H. Tian and Y. Hou, State feedback design for set stabilization of probabilistic Boolean control networks, Journal of the Franklin Institute, 356 (2019), 4358-4377. doi: 10.1016/j.jfranklin.2018.12.027. Google Scholar
[36]	H. Tian, H. Zhang, Z. Wang and Y. Hou, Stabilization of k-valued logical control networks by open-loop control via the reverse-transfer method, Automatica, 83 (2017), 387-390. doi: 10.1016/j.automatica.2016.12.040. Google Scholar
[37]	B. Wang, J. Feng, H. Li and Y. Yu, On detectability of Boolean control networks, Nonlinear Analysis: Hybrid Systems, 36 (2020), 100859, 18 pp. doi: 10.1016/j.nahs.2020.100859. Google Scholar
[38]	S. Wang, J. Feng, Y. Yu and J. Zhao, Further results on dynamic-algebraic Boolean control networks, Science China Information Sciences, 62 (2019), 12208, 14 pp. doi: 10.1007/s11432-018-9447-4. Google Scholar
[39]	M. Xu, Y. Liu, J. Lou, Z. Wu and J. Zhong, Set stabilization of probabilistic Boolean control networks: A sampled-data control approach, IEEE Transactions on Cybernetics, 2019, 1–8. doi: 10.1109/TCYB.2019.2940654. Google Scholar
[40]	D. Yang, X. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305. doi: 10.1016/j.nahs.2019.01.006. Google Scholar
[41]	X. Yang, X. Li, Q. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences and Engineering, 15 (2018), 1495-1515. doi: 10.3934/mbe.2018069. Google Scholar
[42]	Y. Yu, J. Feng, J. Pan and D. Cheng, Block decoupling of Boolean control networks, IEEE Transactions on Automatic Control, 64 (2019), 3129-3140. doi: 10.1109/TAC.2018.2880411. Google Scholar
[43]	Y. Yu, M. Meng, and J. Feng, Observability of Boolean networks via matrix equations, Automatica, 111 (2020), 108621, 5 pp. doi: 10.1016/j.automatica.2019.108621. Google Scholar
[44]	Q. Zhang, J. Feng, J. Pan and J. Xia, Set controllability for switched Boolean control networks, Neurocomputing, 359 (2019), 476-482. doi: 10.1016/j.neucom.2019.05.087. Google Scholar
[45]	R. Zhou, Y. Guo and W. Gui, Set reachability and observability of probabilistic Boolean networks, Automatica, 106 (2019), 230-241. doi: 10.1016/j.automatica.2019.05.021. Google Scholar
[46]	S. Zhu, J. Lu, Y. Liu, T. Huang and J. Kurths, Output tracking of probabilistic Boolean networks by output feedback control, Information Sciences, 483 (2019), 96-105. doi: 10.1016/j.ins.2018.12.087. Google Scholar
[47]	R. Zhou and Y. Guo, Set stabilization in distribution of probabilistic Boolean control networks, in 2018 13th World Congress on Intelligent Control and Automation (WCICA), (2018), 274–279. Google Scholar

Figure 1. Trajectory of $ h(t) $, where $ x(t) = \delta_8^{h(t)} $

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Figure 2. Trajectory of $ h(t) $, where $ x(t) = \delta_8^{h(t)} $

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2019 Impact Factor: 1.233

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2019 Impact Factor: 1.233

American Institute of Mathematical Sciences

State feedback for set stabilization of Markovian jump Boolean control networks

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American Institute of Mathematical Sciences

State feedback for set stabilization of Markovian jump Boolean control networks

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