Geodesic
Invariant sets of control systems with distributed parameters with time delay
Eurasian mathematical journal, Tome 5 (2014) no. 4, pp. 70-78 
U. Ibragimov. Invariant sets of control systems with distributed parameters with time delay. Eurasian mathematical journal, Tome 5 (2014) no. 4, pp. 70-78. http://geodesic.mathdoc.fr/item/EMJ_2014_5_4_a4/
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     author = {U. Ibragimov},
     title = {Invariant sets of control systems with distributed parameters with time delay},
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     volume = {5},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2014_5_4_a4/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider the problem of finding conditions ensuring that a given set is strong or weak invariant with respect to control system with time delay. System is described by heat conductivity equation in right-hand side of which there is the control in additive form. Necessary and sufficient conditions were obtained for the invariance of a given set under a geometric restriction and sufficient conditions under an integral restriction. The given conditions differ from results for the control with time delay obtained earlier.

[1] J.-P. Aubin, “A survey of viability theory”, SIAM J. Contr. and Optim., 28:4 (1990), 749–788 | DOI | MR | Zbl

[2] A. Feuer, J. Heymann, “$\Omega$-invariance in control systems with bounded controls”, J. Math. Anal. and Appl., 53:2 (1976), 266–276 | DOI | MR | Zbl

[3] M. Tukhtasinov, U. M. Ibragimov, “Sets invariant under an integral constraint on controls”, Russian Mathematics, 55:8 (2011), 59–65 | DOI | MR | Zbl

[4] H. T. Banks, Sava Dediu, Hoan K. Nguyen, “Time delay systems with distribution dependent dynamics”, Annual Reviews in Control, 31:1 (2007), 17–26 | DOI

[5] B. Chen, X. P. Liu, S. C. Tong, “Delay-dependent stability analysis and control synthesis of fuzzy dynamic systems with time delay”, Fuzzy Sets Syst., 157:16 (2006), 24–40 | DOI | MR