Theorem on the existence of two-point oscillatory solutions to a relay perturbed system with a negative eigenvalue of the matrix

V. V. Yevstafyeva

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 2, pp. 990-1010

Résumé

We consider a multidimensional system of ordinary first-order differential equations with a nonlinearity of a non-ideal relay and a continuous periodic function of perturbation in the right-hand side. The system matrix has simple, real, nonzero eigenvalues and at least one is negative. We study continuous oscillatory solutions with two switching points in phase space and with the same time of return to each of these points on the discontinuity surface. The theorem of both the existence of the solution and its parameters is established. An example illustrating the obtained theoretical results is given.

Détail

Keywords: $n$-dimensional ODE system, non-ideal relay, positive hysteresis, periodic function of perturbation, bounded solutions, switching points, periodicity on return time, simple real eigenvalues, nonsingular Lurie transformation, system of transcendental equations.

Bibliographie

[1] A.A. Andronov, A.A. Vitt, S.E. Khaĭkin, Theory of oscillators, Pergamon Press, Oxford etc, 1966 | MR | Zbl

[2] B.A. Fleishman, “Harmonic and subharmonic response of an on-off control system to sinusoidal inputs”, J. Franklin Inst., 270:2 (1960), 99–113 | DOI | MR | Zbl

[3] M.A. Krasnosel'skii, A.V. Pokrovskii, “Periodic oscillations in systems with relay nonlinearities”, Sov. Math. Dokl., 15:3 (1974), 873–877 | MR | Zbl

[4] A. Visintin, “P.D.E.s with hysteresis 30 years later”, Discrete Contin. Dyn. Syst. Ser. S, 8:4 (2015), 793–816 | MR | Zbl

[5] A.Yu. Kolesov, Yu.S. Kolesov, Relaxational oscillations in mathematical models of ecology, Proc. Steklov Inst. Math., 199, AMS, Providence, 1995 | MR | Zbl

[6] D.I. Rachinskii, “The Cauchy problem for differential equations with a Mroz hysteresis nonlinearity”, Differ. Equ., 33:8 (1997), 1047–1054 | MR | Zbl

[7] A. Pimenov, D. Rachinskii, “Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator”, Math. Bohem., 139:2 (2014), 285–298 | DOI | MR | Zbl

[8] L. Fang, J. Wang, Q. Zhang, “Identification of extended Hammerstein systems with hysteresis-type input nonlinearities described by Preisach model”, Nonlinear Dyn., 79:2 (2015), 1257–1273 | DOI | Zbl

[9] N.D. Botkin, M. Brokate, E. El Behi-Gornostaeva, “One-phase flow in porous media with hysteresis”, Phys. B: Cond. Matter, 486 (2016), 183–186 | DOI

[10] I.D. Mayergoyz, Mathematical models of hysteresis and their applications, Elsevier, Amsterdam, 2003 | DOI | Zbl

[11] G. Bertotti, I.D. Mayergoyz, The science of hysteresis, Elsevier, Amsterdam, 2005

[12] A. Visintin, “Ten issues about hysteresis”, Acta Appl. Math., 132:1 (2014), 635–647 | DOI | Zbl

[13] V. Hassani, T. Tjahjowidodo, T.N. Do, “A survey on hysteresis modeling, identification and control”, Mech. Syst. Signal Processing, 49:1-2 (2014), 209–233 | DOI

[14] M.A. Krasnosel'skii, A.V. Pokrovskii, Systems with hysteresis, Springer, Berlin etc., 1989 | Zbl

[15] M. Brokate, P. Krejci, “Weak differentiability of scalar hysteresis operators”, Discrete Contin. Dyn. Syst., 35:6 (2015), 2405–2421 | DOI | Zbl

[16] V.I. Utkin, Yu.V. Orlov, “Control systems with vector relays”, Autom. Remote Control, 80:9 (2019), 1671–1680 | DOI | Zbl

[17] C.E.L. da Silva, A. Jacquemard, M.A. Teixeira, “Periodic solutions of a class of non-autonomous discontinuous second-order differential equations”, J. Dyn. Control Syst., 26:1 (2020), 17–44 | DOI | Zbl

[18] A.S. Fursov, R.P. Mitrev, P.A. Krylov, T.S. Todorov, “On the existence of a periodic mode in a nonlinear system”, Differ. Equ., 57:8 (2021), 1076–1087 | DOI | Zbl

[19] M.A. Vasquez-Beltran, B. Jayawardhana, R. Peletier, “Recursive algorithm for the control of output remnant of Preisach hysteresis operator”, IEEE Control Syst. Lett., 5:3 (2021), 1061–1066 | DOI | MR

[20] A.M. Kamachkin, V.V. Yevstafyeva, D.K. Potapov, “The existence of a unique fixed point of mappings generated by a multidimensional system with relay hysteresis”, Differ. Equ., 59:7 (2023), 998–1002 | DOI | Zbl

[21] H. Logemann, E.P. Ryan, “Systems with hysteresis in the feedback loop: Existence, regularity and asymptotic behaviour of solutions”, ESAIM: Control Optim. Calc. Var., 9 (2003), 169–196 | DOI | Zbl

[22] C. Hayashi, “Stability investigation of the nonlinear periodic oscillations”, J. Appl. Phys., 24:3 (1953), 344–348 | DOI | Zbl

[23] Yu.I. Neimark, “Periodic modes and stability in relay systems”, Avtom. Telemekh., 14:5 (1953), 556–569

[24] V.I. Zubov, Oscillations in nonlinear and controlled systems, Sudpromgiz, L., 1962 | Zbl

[25] D.I. Rachinskii, “Asymptotic stability of large-amplitude oscillations in systems with hysteresis”, Nonlinear Differ. Equ. Appl., 6:3 (1999), 267–288 | DOI | Zbl

[26] A.S. Fursov, T.S. Todorov, P.A. Krylov, R.P. Mitrev, “On the existence of oscillatory modes in a nonlinear system with hystereses”, Differ. Equ., 56:8 (2020), 1081–1099 | DOI | Zbl

[27] M.O. Fen, F. Tokmak Fen, “Unpredictable oscillations of SICNNs with delay”, Neurocomputing, 464 (2021), 119–129 | DOI

[28] G.A. Leonov, M.M. Shumafov, V.A. Teshev, K.D. Aleksandrov, “Differential equations with hysteresis operators. Existence of solutions, stability, and oscillations”, Differ. Equ., 53:13 (2017), 1764–1816 | DOI | Zbl

[29] P.-A. Bliman, A.M. Krasnosel'skii, “Periodic solutions of linear systems coupled with relay”, Nonlinear Anal., Theory, Methods Appl., 30:2 (1997), 687–696 | DOI | Zbl

[30] A.M. Solovyov, M.E. Semenov, P.A. Meleshenko, O.O. Reshetova, M.A. Popov, E.G. Kabulova, “Hysteretic nonlinearity and unbounded solutions in oscillating systems”, Procedia Eng., 201 (2017), 578–583 | DOI

[31] M. Brokate, A.V. Pokrovskii, “Asymptotically stable oscillations in systems with hysteresis nonlinearities”, J. Differ. Equ., 150:1 (1998), 98–123 | DOI | Zbl

[32] A.L. Medvedskii, P.A. Meleshenko, V.A. Nesterov, O.O. Reshetova, M.E. Semenov, A.M. Solovyov, “Unstable oscillating systems with hysteresis: problems of stabilization and control”, J. Comput. Syst. Sci. Int., 59:4 (2020), 533–556 | DOI | Zbl

[33] V.V. Yevstafyeva, “On the existence of two-point oscillatory solutions of a perturbed relay system with hysteresis”, Differ. Equ., 57:2 (2021), 155–164 | DOI | Zbl

[34] V.V. Yevstafyeva, “Existence of $T/k$-periodic solutions of a nonlinear nonautonomous system whose matrix has a multiple eigenvalue”, Math. Notes, 109:4 (2021), 551–562 | DOI | Zbl

[35] V.V. Yevstafyeva, “Control design for a perturbed system with an ambiguous nonlinearity”, Autom. Remote Control, 84:3 (2023), 226–239 | DOI | Zbl

[36] A.M. Kamachkin, D.K. Potapov, V.V. Yevstafyeva, “Dynamics of relay systems with hysteresis and harmonic perturbation”, Eurasian Math. J., 15:2 (2024), 48–60 | DOI | Zbl

[37] V.V. Yevstafyeva, A.M. Kamachkin, D.K. Potapov, “Periodic modes in an automatic control system with a three-position hysteresis relay”, Vestn. St. Petersb. Univ. Prikl. Mat. Inform. Prots. Upr., 18:4 (2022), 596–607 | MR

[38] V.V. Yevstafyeva, A.M. Kamachkin, D.K. Potapov, “On one type of oscillating solutions of a second-order ordinary differential equation with a three-position hysteresis relay and perturbation”, Differ. Equ., 59:2 (2023), 153–167 | DOI | Zbl

[39] V.V. Yevstafyeva, “Oscillating solutions of a second-order ordinary differential equation with a three-position hysteresis relay without entering the saturation zones”, Differ. Equ., 59:6 (2023), 726–740 | DOI | Zbl

[40] S. Varigonda, T.T. Georgiou, “Dynamics of relay relaxation oscillators”, IEEE Trans. Autom. Control, 46:1 (2001), 65–77 | DOI | Zbl

[41] V.I. Utkin, Sliding modes and their application in variable structure systems, Mir, M., 1978 | Zbl

[42] A. Jacquemard, M.A. Teixeira, “Periodic solutions of a class of non-autonomous second order differential equations with discontinuous right-hand side”, Physica D: Nonlinear Phenom., 241:22 (2012), 2003–2009 | DOI

[43] M.O. Fen, F. Fen, “Quasilinear systems with unpredictable relay perturbations”, Turk. J. Math., 46:4 (2022), 1369–1383 | DOI | Zbl

[44] Ya.Z. Tsypkin, Relay control systems, Cambridge University Press, Cambridge, 1984 | Zbl

[45] V.V. Yevstafyeva, “Existence of two-point oscillatory solutions of a relay nonautonomous system with multiple eigenvalue of a real symmetric matrix”, Ukr. Math. J., 73:5 (2021), 746–757 | DOI | Zbl

[46] V.V. Yevstafyeva, “Criterion for the existence of two-point oscillatory solution of a perturbed system with a relay”, Math. Notes, 114:2 (2023), 212–222 | DOI | Zbl

[47] V.V. Yevstafyeva, “On one type of oscillatory solutions of a nonautonomous system with relay hysteresis”, Math. Notes, 115:5 (2024), 734–747 | DOI | Zbl

[48] A.M. Kamachkin, D.K. Potapov, V.V. Yevstafyeva, “Two-point oscillatory solutions to system with relay hysteresis and nonperiodic external disturbance”, Appl. Math., Praha, 69:3 (2024), 395–414 | DOI | Zbl

[49] M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, Piecewise-smooth dynamical systems. Theory and applications, Springer, New York, 2008 | Zbl

[50] M. Arnold, N. Begun, P. Gurevich, E. Kwame, H. Lamba, D. Rachinskii, “Dynamics of discrete time systems with a hysteresis stop operator”, SIAM J. Appl. Dyn. Syst., 16:1 (2017), 91–119 | DOI | Zbl

[51] R.A. Nelepin, Exact analytical methods in the theory of nonlinear automatic systems, Sudostroenie, L., 1967 | Zbl

[52] A.M. Kamachkin, V.N. Shamberov, “The decomposition method of research into the nonlinear dynamical systems' space of parameters”, Appl. Math. Sci., 9:81 (2015), 4009–4018 | DOI

[53] D. Rachinskii, “Realization of arbitrary hysteresis by a low-dimensional gradient flow”, Disc. Contin. Dyn. Syst. Ser. B, 21:1 (2016), 227–243 | DOI | Zbl