Stability of two-parameter systems of linear autonomous differential equations with bounded delay

M. V. Mulyukov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 51 (2018), pp. 79-122.

Résumé

We consider a system of linear autonomous differential equations with bounded delay in the case where its characteristic function depends linearly on two scalar parameters. The development of the D-subdivision method is carried out in connection with the problem of constructing the stability domain of this system. Firstly, a complete classification of the points and lines of D-subdivision is carried out. Secondly, a complete classification of two-parameter characteristic equations by the type and structure of D-subdivision domains is carried out. All equations are divided into four types: D-subdivision domains of equations of the first type have curvilinear boundaries, D-subdivision domains of equations of the second and the third type have only rectilinear boundaries, equations of the fourth type are stable or unstable regardless of parameter values. Thirdly, for each type of equations, new methods of selecting the stability domain among regions of D-subdivision are developed. On the basis of the results obtained, stability domains are constructed for certain differential equations and systems of equations with concentrated and distributed delay.

Détail

Keywords: delay differential equations, systems of differential equations, autonomous equations, asymptotic stability, D-subdivision method, stability domain.

Bibliographie

[1] Myshkis A.D., Linear differential equations with delayed argument, Nauka, M., 1972, 351 pp.

[2] Azbelev N.V., Maksimov V.P., Rakhmatullina N.F., Introduction to the theory of functional-differential equations, Nauka, M., 1991, 280 pp.

[3] Azbelev N.V., Simonov P.M., Stability of solutions of the equations with ordinary derivatives, Perm University, Perm, 2001, 230 pp.

[4] Kolmanovskii V.B., Nosov V.R., Stability and periodic regimes of controlled systems with aftereffect, Nauka, M., 1981, 448 pp.

[5] El'sgol'ts L.E., Norkin S.B., Introduction to the theory of differential equations with deviating argument, Nauka, M., 1971, 296 pp.

[6] Hale J.K., Theory of functional-differential equations, Springer-Verlag, New York, 1977, 365 pp. | MR | Zbl

[7] Azbelev N.V., Simonov P.M., “Stability of equations with delay”, Russ. Math., 41:6 (1997), 3–16 | MR | Zbl

[8] Zubov V.I., “On the theory of linear stationary systems with lagging arguments”, Izv. Vyssh. Uchebn. Zaved. Mat., 1958, no. 6, 86–95 (in Russian)

[9] Postnikov M.M., Stable Polynomials, Editorial URSS, M., 2004, 176 pp.

[10] Pontryagin L.S., “On zeros of some transcendental functions”, Izv. Akad. Nauk SSSR Ser. Mat., 6:3 (1942), 115–134 (in Russian)

[11] Chebotarev N.G., Meiman N.N., “The Routh–Hurwitz problem for polynomials and entire functions”, Trudy Mat. Inst. Steklov, 26, 1949, 3–331 (in Russian)

[12] Mikhailov A.V., “The methods of harmonie analysis in the theory of control”, Avtomat. i Telemekh., 1938, no. 3, 27–81 (in Russian)

[13] Neimark Yu.I., “On the determination of the values of the parameters for which a system of automatic regulation is stable”, Avtomat. i Telemekh., 9:3 (1948), 190–203 (in Russian)

[14] Neimark Yu.I., Stability of linearized dystems (discrete and distributed), Leningrad Red Banner Air Force Engineering Academy, L., 1949, 140 pp.

[15] Neimark Yu.I., Dynamical systems and controlled processes, Nauka, M., 1978, 336 pp.

[16] Maxwell J.C., Vyshnegradskii I.A., Stodola A., Theory of automatic control (Linearized problems), The Academy of Sciences of the USSR, M., 1949, 430 pp.

[17] Andronov A.A., Maier A.G., “Simplest linear systems with retardation”, Avtomat. i Telemekh., 7:2–3 (1946), 95–106 (in Russian) | Zbl

[18] Levitskaya I.S., “The stability region of a linear differential equation with two delays”, Izvestiya Chelyabinskogo Nauchnogo Tsentra, 23:2 (2004), 7–12 (in Russian)

[19] Kipnis M.M., Levitskaya I.S., “Stability of delay difference and differential equations: similarities and distinctions”, Proc. Internat. Conf. Difference Equations, Special Functions and Orthogonal Polinomials (Munich, Germany, 2005), World Scientific, New Jersey, 2007, 315–324 | DOI | MR | Zbl

[20] Mahaffy J.M., Busken T.C., “Regions of stability for a linear differential equation with two rationally dependent delays”, Discrete and Continuous Dynamical Systems, 35:10 (2015), 4955–4986 | DOI | MR | Zbl

[21] Mulyukov M.V., “Classification of two-parameter autonomous linear systems with delay”, Journal of Mathematical Sciences, 230:5 (2018), 724–727 | DOI | Zbl

[22] Mulyukov M.V., “Structure of the $D$-subdivision domains for the two-parameter characteristic equations of systems with delay”, Functional-differential equiations: theory and applications, Proc. of the conference dedicated to the 95th anniversary of the professor N.V. Azbelev, Perm National Research Polytechnic University, Perm, 2018, 180-200 (in Russian) | Zbl

[23] Mulyukov M.V., “$D$-subdivision domains with straight bounds”, Mathematics in the modern world, Abstracts of the all-Russian conference, Sobolev Institute of Mathematics, Novosibirsk, 2017, 233 (in Russian)

[24] Lankaster P., The theory of matrices, Academic Press, New York, 1969, 316 pp. | MR

[25] Markushevich A.I., Entire functions. Elementary essay, Nauka, M., 1975, 120 pp.

[26] Shabat B.V., Introduction to complex analysis, Textbook for universities, v. 1, Function of one variable, Nauka, M., 1985, 336 pp.

[27] Stepan G., Retarded dynamical systems: stability and characteristic functions, Longman Scientific Technical, 1989, 151 pp. | MR | Zbl

[28] Trenogin V.A., Functional analysis, Fizmatlit, M., 2007, 488 pp.

[29] Sabatullina T.L., Malygina V.V., “Several stability tests for linear autonomous differential equations with distributed delay”, Russian Mathematics, 51:6 (2007), 52–60 | DOI | MR | Zbl

[30] Mulyukov M.V., “Asymptotic stability of two-parameter systems of delay differential equations”, Russian Mathematics, 58:6 (2014), 44–50 | DOI | MR | Zbl

[31] Hsu C.S., Bhatt S.J., “Stability criteria for second-order dynamical systems with time lag”, J. Appl. Mech., 33:1 (1966), 113–118 | DOI | MR | Zbl

[32] Hsu C.S., Bhatt S.J., “Stability charts for second-order dynamical systems with time lag”, J. Appl. Mech., 33:1 (1966), 119–124 | DOI | MR | Zbl

[33] Balandin A.S., Sabatulina T.L., “The local stability of a population dynamics model in conditions of deleterious effects”, Siberian Electronic Mathematical Reports, 12 (2015), 610–624 (in Russian) | Zbl

[34] Mulyukov M.V., “Stability of a linear differential equation with a double delay”, Modern methods of applied mathematics, control theory and computer technologies, Proc. of the VIII international conference, Izdatel'stvo “Nauchnaya kniga”, Voronezh, 2015, 258–260 (in Russian) | Zbl

[35] Mulyukov M.V., “The stability of the linear autonomous differential equation with distributed and concentrated delay”, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 20:5 (2015), 1325–1331 (in Russian)