Geodesic
Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems
Trudy Instituta matematiki i mehaniki, Control, stability, and inverse problems of dynamics, Tome 12 (2006) no. 2, pp. 78-87 Cet article a éte moissonné depuis la source Math-Net.Ru

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The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given. 
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Yu. F. Dolgii. Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems. Trudy Instituta matematiki i mehaniki, Control, stability, and inverse problems of dynamics, Tome 12 (2006) no. 2, pp. 78-87. http://geodesic.mathdoc.fr/item/TIMM_2006_12_2_a6/

[1] Krasovskii N. N., Nekotorye zadachi teorii ustoichivosti dvizheniya, Fizmatgiz, M., 1959 | MR

[2] Shimanov S. N., “K teorii lineinykh differentsialnykh uravnenii s periodicheskimi koeffitsientami i zapazdyvaniem vremeni”, Prikl. matematika i mekhanika, 27:3 (1963), 450–458 | MR | Zbl

[3] Borisovich Yu. G., Subbotin V. F., “Operator sdviga po traektoriyam evolyutsionnykh uravnenii i periodicheskie resheniya”, Dokl. AN SSSR, 175:5 (1967), 9–12 | MR | Zbl

[4] Kleimenov A. F., “Suschestvovanie i ustoichivost periodicheskikh reshenii sistem s zapazdyvaniem, blizkikh k sistemam Lyapunova”, Differents. uravneniya, 4:8 (1968), 1431–1440

[5] Kolesov Yu. S., Shvitra D. I., Avtokolebaniya v sistemakh s zapazdyvaniem, Mokslas, Vilnyus, 1979 | MR | Zbl

[6] Zubov V. I., Kolebaniya v nelineinykh i upravlyaemykh sistemakh, Sudpromizdat, L., 1962 | MR

[7] Kolmanovskii V. B., Nosov V. R., Ustoichivost i periodicheskie rezhimy reguliruemykh sistem s posledeistviem, Nauka, M., 1981 | MR

[8] Mitropolskii Yu. A., Martynyuk D. I., Periodicheskie i kvaziperiodicheskie kolebaniya sistem s zapazdyvaniem, Vischa shkola, Kiev, 1979 | MR

[9] Kheil Dzh., Teoriya funktsionalno-differentsialnykh uravnenii, Mir, M., 1984 | MR

[10] Shimanov S. N., “Nekotorye voprosy teorii kolebanii sistem s zapazdyvaniem”, Pyataya letnyaya mat. shkola, In-t matematiki AN USSR, Kiev, 1968, 473–549 | Zbl

[11] Dormayer P., “The stability of special symmetric solutions $\dot x(t)=\alpha f(x(t-1))$ with small amplitudes”, Nonlinear Analysis, Methods and Applications, 14:8 (1990), 701–715 | DOI | MR | Zbl

[12] Dolgii Yu. F., Nikolaev S. G., “Ustoichivost periodicheskogo resheniya nelineinogo differentsialnogo uravneniya s zapazdyvaniem”, Differents. uravneniya, 37:5 (2001), 592–601 | MR

[13] Dolgii Yu. F., Zakharov A. V., “A delay effect upon periodic oscillations in a conservative system”, Proc. of the Steklov Inst. of Math., Suppl. 2, 2003, S24–S44 | MR

[14] Dolgii Yu. F., Nidchenko S. N., “Bifurkatsionnyi metod issledovaniya ustoichivosti resheniya differentsialnogo uravneniya s zapazdyvaniem”, Sib. mat. zhurn., 46:6 (2005), 1288–1301 | MR | Zbl

[15] Gasilov G. L., “O kharakteristicheskom uravnenii sistemy lineinykh differentsialnykh uravnenii s periodicheskimi koeffitsientami i zapazdyvaniem”, Izv. vuzov. Matematika, 1972, no. 4, 60–66 | MR | Zbl

[16] Zverkin A. M., “K teorii differentsialno-raznostnykh uravnenii s zapazdyvaniyami, soizmerimymi s periodom koeffitsientov”, Differents. uravneniya, 24:9 (1988), 1481–1492 | MR | Zbl

[17] Dolgii Yu. F., Ustoichivost periodicheskikh differentsialno-raznostnykh uravnenii, UrGU, Ekaterinburg, 1996

[18] Dolgii Yu. F., “Neustoichivost analoga uravneniya Khilla s zapazdyvaniem”, Ustoichivost i nelineinye kolebaniya, UrGU, Sverdlovsk, 1984, 30–36

[19] Dolgii Yu. F., “Neustoichivost analoga gamiltonovoi sistemy s zapazdyvaniem”, Ustoichivost i nelineinye kolebaniya, UrGU, Sverdlovsk, 1986, 13–21

[20] Dolgii Yu. F., Shimanov S. N., “Suschestvovanie zony ustoichivosti dlya odnogo uravneniya s zapazdyvaniem”, Ustoichivost i nelineinye kolebaniya, UrGU, Sverdlovsk, 1988, 11–18

[21] Dolgii Yu. F., “Ustoichivost odnoi periodicheskoi sistemy s postoyannym zapazdyvaniem”, Funktsionalno-differentsialnye uravneniya, PPI, Perm, 1988, 131–136 | MR

[22] Dolgii Yu. F., “Ob ustoichivosti odnoi periodicheskoi sistemy s zapazdyvaniem”, Kraevye zadachi, PPI, Perm, 1989, 16–21 | MR

[23] Dolgii Yu. F., Nikolaev S. G., “Neustoichivost odnoi periodicheskoi sistemy s zapazdyvaniem”, Differents. uravneniya, 34:4 (1998), 465–470 | MR

[24] Dolgii Yu. F., Nikolaev S. G., “Ob ustoichivosti periodicheskoi sistemy differentsialnykh uravnenii s zapazdyvaniem”, Differents. uravneniya, 35:10 (1999), 1330–1336 | MR | Zbl

[25] Yakubovich V. A., Starzhinskii V. M., Lineinye differentsialnye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya, Nauka, M., 1972 | MR

[26] Gantmakher F. R., Teoriya matrits, Nauka, M., 1988 | MR | Zbl

[27] Krein M. G., “O priznakakh ustoichivoi ogranichennosti reshenii periodicheskikh kanonicheskikh sistem”, Prikl. matematika i mekhanika, 19:6 (1955), 641–680 | MR

[28] Nikolaev S. G., Ustoichivost lineinykh periodicheskikh sistem s postoyannym zapazdyvaniem, Diss. $\dots$ kand. fiz.-mat. nauk, IMM UrO RAN, Ekaterinburg, 2000