In this study, implicit Boolean networks (IBNs), which are more general than classic BNs, are proposed for the first time motivated by the river-crossing decision problem. By resorting to the admissible set, some necessary and sufficient conditions are established, under which IBNs can be equivalently converted into classic BNs or restricted BNs. Subsequently, an improved approach is presented to determine the topological structure of dynamic-algebraic BNs (D-ABNs), based on which transformation relations between IBNs and D-ABNs are given. Finally, a biochemical oscillator example is used to show the application of the obtained results.

References

1. 1)
  - 9. Kauffman, S.A.: ‘Metabolic stability and epigenesis in randomly constructed genetic nets’, J. Theor. Biol., 1969, 22, (3), pp. 437–467.
2. 2)
  - 41. Li, H., Wang, Y.: ‘Stability analysis for switched singular Boolean networks’, Control Theory Appl., 2014, 31, pp. 908–914.
3. 3)
  - 10. Cheng, D., Qi, H., Liu, T., et al: ‘A note on observability of Boolean control networks’, Syst. Control Lett., 2016, 87, pp. 76–82.
4. 4)
  - 37. Ito, H., Langerman, S., Yoshida, Y.: ‘Generalized river crossing problems’, Theory Comput. Syst., 2015, 56, (2), pp. 418–435.
5. 5)
  - 40. Yu, Y., Feng, J., Wang, S.: ‘Explicit formula of logical algebraic equations and singular Boolean networks with probability’. Proc. 35th Chinese Control Conf., Chengdu, China, July 2016, pp. 1192–1197.
6. 6)
  - 34. Li, H., Wang, Y., Xie, L., et al: ‘Disturbance decoupling control design for switched Boolean control networks’, Syst. Control Lett., 2014, 72, (72), pp. 1–6.
7. 7)
  - 36. Devdariani, E.N., Ledyaev, Y.S.: ‘Maximum principle for implicit control systems’, Appl. Math. Optim., 1999, 40, (1), pp. 79–103.
8. 8)
  - 39. Kim, K.: ‘Boolean matrix theory and applications’ (Dekker Press, 1982).
9. 9)
  - 16. Yang, M., Li, R., Chu, T.: ‘Controller design for disturbance decoupling of Boolean control networks’, Automatica, 2013, 49, (1), pp. 273–277.
10. 10)
  - 38. Ding, Z., Jiang, C.: ‘Application of temporal Petri nets in intelligent control’, Control Decis., 2002, 17, (2), pp. 230–232.
11. 11)
  - 6. Wang, Y., Zhang, C., Liu, Z.: ‘A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems’, Automatica, 2012, 48, (7), pp. 1227–1236.
12. 12)
  - 15. Cheng, D.: ‘Disturbance decoupling of Boolean control networks’, IEEE Trans. Autom. Control, 2011, 56, (1), pp. 2–10.
13. 13)
  - 30. Meng, M., Feng, J.: ‘Optimal control problem of singular Boolean control networks’, Int. J. Control Autom. Syst., 2015, 13, (2), pp. 266–273.
14. 14)
  - 18. Li, F., Sun, J.: ‘Stability and stabilization of Boolean networks with impulsive effects’, Syst. Control Lett., 2012, 61, (1), pp. 1–5.
15. 15)
  - 35. Li, H., Wang, Y.: ‘Controllability analysis and control design for switched Boolean networks with state and input constraints’, SIAM J. Control Optim., 2015, 53, (5), pp. 2955–2979.
16. 16)
  - 8. Cheng, D., Qi, H.: ‘Controllability and observability of Boolean control networks’, Automatica, 2009, 45, (7), pp. 1659–1667.
17. 17)
  - 23. Liu, Y., Sun, L., Lu, J., et al: ‘Feedback controller design for the synchronization of Boolean control networks’, IEEE Trans. Neural Netw. Learn. Syst., 2015, 27, (9), pp. 1991–1996.
18. 18)
  - 1. Cheng, D., Qi, H., Li, Z.: ‘Analysis and control of Boolean networks: a semi-tensor product approach’ (Springer London Press, 2011).
19. 19)
  - 28. Meng, M., Feng, J.: ‘Topological structure and the disturbance decoupling problem of singular Boolean networks’, IET Control Theory Appl., 2014, 8, (13), pp. 1247–1255.
20. 20)
  - 25. Meng, M., Feng, J.: ‘Function perturbations in Boolean networks with its application in D. melanogaster gene network’, Eur. J. Control, 2014, 20, (2), pp. 87–94.
21. 21)
  - 21. Cheng, D., Zhao, Y.: ‘Identification of Boolean control networks’, Automatica, 2011, 47, (4), pp. 702–710.
22. 22)
  - 14. Liu, Y., Chen, H., Lu, J., et al: ‘Controllability of probabilistic Boolean control networks based on transition probability matrices’, Automatica, 2015, 52, pp. 340–345.
23. 23)
  - 20. Cheng, D., Li, Z., Qi, H.: ‘Realization of Boolean control networks’, Automatica, 2010, 46, (1), pp. 62–69.
24. 24)
  - 5. Guo, P., Wang, Y., Li, H.: ‘Algebraic formulation and strategy optimization for a class of evolutionary networked games via semi-tensor product method’, Automatica, 2013, 49, (11), pp. 3384–3389.
25. 25)
  - 24. Guo, Y., Wang, P., Gui, W., et al: ‘Set stability and set stabilization of Boolean control networks based on invariant subsets’, Automatica, 2015, 61, pp. 106–112.
26. 26)
  - 17. Cheng, D., Qi, H., Li, Z., et al: ‘Stability and stabilization of Boolean networks’, Int. J. Robust Nonlinear Control, 2011, 21, (2), pp. 134–156.
27. 27)
  - 22. Cheng, D., Qi, H., Li, Z.: ‘Model construction of Boolean network via observed data’, IEEE Trans. Neural Netw., 2011, 22, (4), pp. 525–536.
28. 28)
  - 32. Li, H., Wang, Y.: ‘On reachability and controllability of switched Boolean control networks’, Automatica, 2012, 48, (11), pp. 2917–2922.
29. 29)
  - 13. Liu, Y., Chen, H., Wu, B.: ‘Controllability of Boolean control networks with impulsive effects and forbidden states’, Math. Methods Appl. Sci., 2014, 37, (1), pp. 1–9.
30. 30)
  - 31. Shmulevich, I., Dougherty, E.R., Kim, S., et al: ‘Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks’, IEEE/ACM Trans. Comput. Biol. Bioinf., 2002, 18, (2), pp. 261–274.
31. 31)
  - 12. Laschov, D., Margaliot, M.: ‘Controllability of Boolean control networks via the Perron–Frobenius theory’, Automatica, 2012, 48, (6), pp. 1218–1223.
32. 32)
  - 27. Feng, J., Yao, J., Cui, P.: ‘Singular Boolean networks: semi-tensor product approach’, Sci. China Inf. Sci., 2013, 56, (11), pp. 1–14.
33. 33)
  - 11. Laschov, D., Margaliot, M., Even, G.: ‘Observability of Boolean networks: a graph-theoretic approach’, Automatica, 2013, 49, (8), pp. 2351–2362.
34. 34)
  - 7. Cheng, D., Qi, H.: ‘A linear representation of dynamics of Boolean networks’, IEEE Trans. Autom. Control, 2010, 55, (10), pp. 2251–2258.
35. 35)
  - 2. Cheng, D., Qi, H., Zhao, Y.: ‘An introduction to analysis and control of Boolean networks: a semi-tensor product of matrices and its applications’ (World Scientific Press, 2012).
36. 36)
  - 4. Cheng, D., He, F., Qi, H., et al: ‘Modeling, analysis and control of networked evolutionary games’, IEEE Trans. Autom. Control, 2015, 60, (9), pp. 2402–2415.
37. 37)
  - 29. Meng, M., Li, B., Feng, J.: ‘Controllability and observability of singular Boolean control networks’, Circuits Syst. Signal Process., 2015, 34, (4), pp. 1233–1248.
38. 38)
  - 3. Cheng, D.: ‘On finite potential games’, Automatica, 2014, 50, (7), pp. 1793–1801.
39. 39)
  - 26. Cheng, D., Zhao, Y., Xu, X.: ‘Mix-valued logic and its applications’, J. Shandong Univ., 2011, 46, (10), pp. 32–44.
40. 40)
  - 33. Li, H., Wang, Y.: ‘Consistent stabilizability of switched Boolean networks’, IEEE Trans. Neural Netw., 2013, 46, pp. 183–189.
41. 41)
  - 19. Liu, Y., Cao, J., Sun, L., et al: ‘Sampled-data state feedback stabilization of Boolean control networks’, Neural Comput., 2016, 28, (4), pp. 778–7995.

Topological structure of implicit Boolean networks

References

Related content