Geodesic
Controllability of linear impulsive matrix Lyapunov differential systems with delays in the control function
Kybernetika, Tome 54 (2018) no. 4, pp. 664-698 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper, we establish the controllability conditions for a finite-dimensional dynamical control system modelled by a linear impulsive matrix Lyapunov ordinary differential equations having multiple constant time-delays in control for certain classes of admissible control functions. We characterize the controllability property of the system in terms of matrix rank conditions and are easy to verify. The obtained results are applicable for both autonomous (time-invariant) and non-autonomous (time-variant) systems. Two numerical examples are given to illustrate the theoretical results obtained in this paper.
In this paper, we establish the controllability conditions for a finite-dimensional dynamical control system modelled by a linear impulsive matrix Lyapunov ordinary differential equations having multiple constant time-delays in control for certain classes of admissible control functions. We characterize the controllability property of the system in terms of matrix rank conditions and are easy to verify. The obtained results are applicable for both autonomous (time-invariant) and non-autonomous (time-variant) systems. Two numerical examples are given to illustrate the theoretical results obtained in this paper.
DOI : 10.14736/kyb-2018-4-0664
Classification : 34A37, 93B05, 93C05, 93C15
Keywords: matrix Lyapunov systems; controllability; impulsive differential systems; delays 
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S. Muni, Vijayakumar; K. George, Raju. Controllability of linear impulsive matrix Lyapunov differential systems with delays in the control function. Kybernetika, Tome 54 (2018) no. 4, pp. 664-698. doi: 10.14736/kyb-2018-4-0664

[1] Balachandran, K., Somasundaram, D.: Controllability of a class of nonlinear systems with distributed delays in control. Kybernetika 19 (1983), 475-482. | MR

[2] Balachandran, K., Somasundaram, D.: Controllability of nonlinear systems consisting of a bilinear mode with time-varying delays in control. Automatica 20 (1984), 257-258. | DOI | MR

[3] Balachandran, K., Somasundaram, D.: Relative controllability of nonlinear systems with time-varying delays in control. Kybernetika 21 (1985), 65-72. | MR

[4] Benzaid, Z., Sznaier, M.: Constrained controllability of linear impulsive differential systems. IEEE Trans. Automat. Control 39 (1994), 1064-1066. | DOI | MR

[5] Bian, W.\.M.: Constrained controllability of some nonlinear systems. Appl. Anal. 72 (1999), 57-73. | DOI | MR

[6] Chyung, D. H.: Controllability of linear time-varying systems with delays. IEEE Trans. Automat. Control 16 (1971), 493-495. | DOI | MR

[7] Dacka, C.: Relative controllability of perturbed nonlinear systems with delay in control. IEEE Trans. Automat. Control 27 (1982), 268-270. | DOI | MR

[8] Dubey, B., George, R. K.: Controllability of semilinear matrix Lyapunov systems. EJDE 42 (2013), 1-12. | MR

[9] Dubey, B., George, R. K.: Controllability of impulsive matrix Lyapunov systems. Appl. Math. Comput. 254 (2015), 327-339. | DOI | MR

[10] Erneux, T.: Applied Delay Differential Equations. Springer-Verlag, New York, USA 2009. | MR

[11] George, R. K., Nandakumaran, A. K., Arapostathis, A.: A note on controllability of impulsive systems. J. Math. Anal. Appl. 241 (2000), 276-283. | DOI | MR

[12] Graham, A.: Kronecker Products and Matrix Calculus: With Applications. Ellis Horwood Ltd. England 1981. | MR

[13] Guan, Z. H., Qian, T. H., Yu, X.: On controllability and observability for a class of impulsive systems. Syst. Control Lett. 47 (2002), 247-257. | DOI | MR

[14] 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 (2002), 1198-1208. | DOI | MR

[15] 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 (2012), 1867-1870. | DOI | MR

[16] Klamka, J.: Relative controllability and minimum energy control of linear systems with distributed delays in control. IEEE Trans. Automat. Control 21 (1976), 594-595. | DOI | MR

[17] Klamka, J.: Controllability of linear systems with time-variable delays in control. Int. J. Control 24 (1976), 869-878. | DOI | MR

[18] Klamka, J.: Absolute controllability of linear systems with time-variable delays in control. Int. J. Control 26 (1977), 57-63. | DOI | MR

[19] Klamka, J.: On the controllability of linear systems with delays in the control. Int. J. Control 25 (1977), 875-883. | DOI | MR

[20] Klamka, J.: Controllability of non-linear systems with distributed delays in control. Int. J. Control 31 (1980), 811-819. | DOI | MR

[21] Klamka, J.: Constrained controllability of nonlinear systems. J. Math. Anal. Appl. 201 (1996), 365-374. | DOI | MR

[22] Klamka, J.: Constrained controllability of semilinear systems with multiple delays in control. Bull. Pol. Ac. Tech. 52 (2004), 25-30. | DOI | MR

[23] Klamka, J.: Constrained controllability of semilinear systems with delayed controls. Bull. Pol. Ac. Tech. 56 (2008), 333-337. | MR

[24] Klamka, J.: Constrained controllability of semilinear systems with delays. Nonlin. Dyn. 56 (2009), 169-177. | DOI | MR

[25] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of impulsive differential equations. World Scientific, Singapore 1989. | MR

[26] Leela, S., McRae, F. A., Sivasundaram, S.: Controllability of impulsive differential equations. J. Math. Anal. Appl. 177 (1993), 24-30. | DOI | MR

[27] Murty, M. S. N., Rao, B. V. Appa, Kumar, G. Suresh: Controllability, observability and realizability of matrix Lyapunov systems. B. Korean Math. Soc. 43 (2006), 149-159. | DOI | MR

[28] Olbrot, A. W.: On controllability of linear systems with time-delays in control. IEEE Trans. Automat. Control 17 (1972), 664-666. | DOI | MR

[29] Sakthivel, R., Mahmudov, N. I., Kim, J. H.: Approximate controllability of nonlinear impulsive differential systems. Rep. Math. Phys. 60 (2007), 85-96. | DOI | MR

[30] Sebakhy, O., Bayoumi, M. M.: A simplified criterion for the controllability of linear systems with delay in control. IEEE Trans. Automat. Control 16 (1971), 364-365. | DOI | MR

[31] Somasundaram, D., Balachandran, K.: Controllability of nonlinear systems consisting of a bilinear mode with distributed delays in control. IEEE Trans. Automat. Control 29 (1984), 573-575. | DOI | MR

[32] Xie, G., Wang, L.: Controllability and observability of a class of linear impulsive systems. J. Math. Anal. Appl. 304 (2005), 336-355. | DOI | MR

[33] Zhao, S., Sun, J.: Controllability and observability for a class of time-varying impulsive systems. Nonlin. Anal. RWA. 10 (2009), 1370-1380. | DOI | MR

[34] Zhu, Z. Q., Lin, Q. W.: Exact controllability of semilinear systems with impulses. Bull. Math. Anal. Appl. 4 (2012), 157-167. | MR

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