Abstract
This article proposes various sufficient controllability criteria for a class of networked higher dimensional systems under the influence of impulses exhibited by their state functions. The conditions obtained are characterised in terms of impulse matrices, inner coupling matrix, network topology and the system matrices. It is demonstrated that Kalman’s rank condition and Popov–Belevitch–Hautus (PBH)-rank condition are just sufficient conditions for the controllability of these systems, but not necessary unlike as that of the networked systems without impulses. Various numerical examples are provided to validate the theoretical results. Further, the control function and controlled trajectory are plotted for the systems considered, that helps to estimate the cost of controller.
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References
Cowan, N.J., Chastain, E.J., Vilhena, D.A., Freudenberg, J.S., Bergstrom, C.T.: Nodal dynamics, not degree distributions, determine the structural controllability of complex networks. PLoS One (2012).https://doi.org/10.1371/journal.pone.0038398
Diblík, J., Khusainov, D.Y., R\(\mathring{{\rm u}}\)žičková, M.: Controllability of linear discrete systems with constant coefficients and pure delay. SIAM J. Control Optim. 47(3), 1140–1149 (2008).https://doi.org/10.1137/070689085
Diblík, J.: Relative and trajectory controllability of linear discrete systems with constant coefficients and a single delay. IEEE Trans. Autom. Control. 64(5), 2158–2165 (2019). https://doi.org/10.1109/TAC.2018.2866453
Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks: From biological nets to the internet and WWW. Oxford University Press, Oxford (2003)
George, R.K., Nandakumaran, A.K., Arapostathis, A.: A note on controllability of impulsive systems. J. Math. Anal. Appl. 241(2), 276–283 (2000). https://doi.org/10.1006/jmaa.1999.6632
Guan, Z.H., Qian, T.H., Yu, X.: Controllability and observability of linear time-varying impulsive systems. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(8), 1198–1208 (2002). https://doi.org/10.1109/TCSI.2002.801261
Guan, Z.H., Qian, T.H., Yu, X.: On controllability and observability for a class of impulsive systems. Syst. Control Lett. 47(3), 247–257 (2002). https://doi.org/10.1016/S0167-6911(02)00204-9
Guo, M., Xue, X., Li, R.: Controllability of impulsive evolution inclusions with nonlocal conditions. J. Optim. Theory Appl. 120(2), 355–374 (2004). https://doi.org/10.1023/B:JOTA.0000015688.53162.eb
Han, J., Liu, Y., Zhao, S., Yang, R.: A note on the controllability and observability for piecewise linear time-varying impulsive systems. Asian J. Control. 15(6), 1867–1870 (2013). https://doi.org/10.1002/asjc.642
Leela, S., McRae, F.A., Sivasundaram, S.: Controllability of impulsive differential equations. J. Math. Anal. Appl. 177(1), 24–30 (1993). https://doi.org/10.1006/jmaa.1993.1240
Leiva, H.: Controllability of semilinear impulsive nonautonomous systems. Int. J. Control 88(3), 585–592 (2015). https://doi.org/10.1080/00207179.2014.966759
Liu, X.: Impulsive control and optimization. Appl. Math. Comput. 73(1), 77–98 (1995). https://doi.org/10.1016/0096-3003(94)00204-H
Liu, Y.Y., Slotine, J.J., Barabási, A.L.: Controllability of complex networks. Nature 473(7346), 167–173 (2011). https://doi.org/10.1038/nature10011
Nieto, J.J., Tisdell, C.C.: On exact controllability of first-order impulsive differential equations. Adv. Differ. Equations 136504, 1–9 (2010). https://doi.org/10.1155/2010/136504
Terrell, W.J.: Stability and stabilization: An introduction. Princeton University Press, Princeton (2009)
Wang, W.X., Ni, X., Lai, Y.C., Grebogi, C.: Optimizing controllability of complex networks by minimum structural perturbations. Phys. Rev. E. (2012). https://doi.org/10.1103/PhysRevE.85.026115
Wang, L., Chen, G., Wang, X., Tang, W.K.S.: Controllability of networked MIMO systems. Automatica 69, 405–409 (2016). https://doi.org/10.1016/j.automatica.2016.03.013
Wang, L., Wang, X., Chen, G.: Controllability of networked higher-dimensional systems with one-dimensional communication. Philos. Trans. A. (2017). https://doi.org/10.1098/rsta.2016.0215
Zhao, S., Sun, J.: Controllability and observability for a class of time-varying impulsive systems. Nonlinear Anal. RWA 10(3), 1370–1380 (2009). https://doi.org/10.1016/j.nonrwa.2008.01.012
Zhao, S., Sun, J.: Controllability and observability for impulsive systems in complex fields. Nonlinear Anal. RWA 11(3), 1513–1521 (2010). https://doi.org/10.1016/j.nonrwa.2009.03.009
Zhou, T.: On the controllability and observability of networked dynamic systems. Automatica 52, 63–75 (2015). https://doi.org/10.1016/j.automatica.2014.10.121
Zhu, Z.Q., Lin, Q.W.: Exact controllability of semilinear systems with impulses. Bull. Math. Anal. Appl. 4(1), 157–167 (2012)
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Muni, V.S., George, R.K. On Controllability of Networked Higher Dimensional Impulsive Systems. Bull. Iran. Math. Soc. 47, 1947–1968 (2021). https://doi.org/10.1007/s41980-020-00481-8
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DOI: https://doi.org/10.1007/s41980-020-00481-8