Geodesic
Methods for studying the stability and stabilization of some systems with large delay
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 15 (2022) no. 4, pp. 99-108 Cet article a éte moissonné depuis la source Math-Net.Ru

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The article is devoted to the study of the properties of systems of differential equations containing a large (in particular, linear) delay. Systems with linear delay have a fairly wide application in biology, in particular, in modelling the distribution of cells in body tissues, as well as in the theory of neural networks. Equations of this type are also found in problems of physics and mechanics, where an important point is the asymptotic behavior of the solution (in particular, the asymptotic stability). When such systems are unstable, the problem of stabilization arises. The optimal stabilization algorithm is based on an union of stabilization of systems of ordinary differential equations and further difference systems. This algorithm is quite simply implemented using numerical methods for solving systems of differential equations with a delay and solving matrix equations. We developed a program that allows quite effectively find a control effect that stabilizes some systems.
Keywords: delay, stability, stabilization. 
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B. G. Grebenshchikov; A. B. Lozhnikov. Methods for studying the stability and stabilization of some systems with large delay. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 15 (2022) no. 4, pp. 99-108. http://geodesic.mathdoc.fr/item/VYURU_2022_15_4_a8/

[1] Krasovskii N.N., Some Problems in the Theory of Stability of Motion, Fizmatgiz, M., 1959, 211 pp. (in Russian)

[2] Shimanov S.N., “On the Instability of the Movement of Systems with Time Delays”, Applied Mathematics and Mechanics, 24:1 (1960), 55–63 (in Russian)

[3] Grebenshchikov B.G., Rozhkov V.I., “Asymptotic Behavior of the Solution of a Stationary System with Delay”, Differential Equations, 29:5 (1993), 640–647 | MR | MR

[4] Grebenshchikov B.G., Novikov S.I., “On the Instability of a System with Linear Delay that is Reducible to a Singularly Perturbed System”, Russian Mathematics, 54:2 (2010), 1–10 | DOI | MR

[5] Repin Yu.M., “On the Conditions of Stability of Systems of Linear Differential Equations with Delays”, Uchenye zapiski Ural'skogo gosudarstvennogo universiteta, 1960, no. 23, 31–34 (in Russian)

[6] Furasov V.D., Stability and Stabilization of Discrete Processes, Nauka, M., 1982, 192 pp. (in Russian) | MR

[7] Furasov V.D., Stability of Motion, Estimation and Stabilization, Nauka, M., 1977 (in Russian) | MR

[8] Kim A.V., Lozhnikov A.B., “A Linear-Quadratic Problem for Systems with Delay in the State. Exact Solutions of Riccati Equations”, Automation and Remote Control, 61:7 (2000), 1076–1090 | MR

[9] Marchenko V.M., “On the Theory of Canonical Forms of Control Systems with Delay”, Matematicheskiy sbornik, 105:3 (1978), 403–412 (in Russian) | MR

[10] Zubov V.I., Lectures on Control Theory, Nauka, M., 1975

[11] Rozhkov V.I., Popov A.M., “Estimates of Solutions of Some Systems of Differential Equations with a Large Delay”, Differential Equations, 7:2 (1971), 271–278 (in Russian) | MR

[12] Grebenshchikov B.G., “Stability with Respect to the First Approximation of Systems with Linearly Time-Dependent Delay”, Differential Equations, 26:2 (1990), 159–162 | MR