On Necessary Conditions for Global Asymptotic Stability of Equilibrium for the Li\'enard Equation

A. O. Ignatyev ; V. V. Kirichenko

Matematičeskie zametki, Tome 93 (2013) no. 1, pp. 63-71.

Résumé

In [1], necessary and sufficient conditions for the global asymptotic stability of the trivial solution of the Liénard equation $\ddot x+f(x)\dot x+g(x)=0$, $g(0)=0$, were obtained under the condition \begin{equation} x\int_0^xf(s)\,ds\geqslant 0 \tag{A} \end{equation} In [1], the following problem was also posed: To determine whether condition (A) is a necessary condition for the global asymptotic stability of the trivial solution of the Liénard equation. The present paper answers this question, and the answer is negative, i.e., condition (A) is not a necessary condition.

Détail

Keywords: Lienard differential equation, global asymptotic stability.

Bibliographie

[1] Ch. Qian, “On global asymptotic stability of second order nonlinear differential systems”, Nonlinear Anal., 22:7 (1994), 823–833 | DOI | MR | Zbl

[2] N. N. Barbashin, E. A. Krasovskii, “Ob ustoichivosti dvizheniya v tselom”, DAN SSSR, 86:6 (1952), 453–456 | MR | Zbl

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

[4] A. O. Ignatev, “Nekotorye obobscheniya teoremy Barbashina–Krasovskogo”, Matem. fizika, 34 (1983), 19–22 | MR | Zbl

[5] A. O. Ignatev, “Ob ustoichivosti pochti periodicheskikh sistem otnositelno chasti peremennykh”, Differents. uravneniya, 25:8 (1989), 1446–1448 | MR

[6] A. O. Ignatyev, “On the stability of equilibrium for almost periodic systems”, Nonlinear Anal., 29:8 (1997), 957–962 | DOI | MR | Zbl

[7] A. O. Ignatyev, “On the asymptotic stability in functional differential equations”, Proc. Amer. Math. Soc., 127:6 (1999), 1753–1760 | DOI | MR | Zbl

[8] A. O. Ignatyev, “On the partial equiasymptotic stability in functional differential equations”, J. Math. Anal. Appl., 268:2 (2002), 615–628 | DOI | MR | Zbl

[9] A. S. Andreev, “Ob asimptoticheskoi ustoichivosti i neustoichivosti nulevogo resheniya neavtonomnogo funktsionalno-differentsialnogo uravneniya”, Problemy analiticheskoi mekhaniki, teorii ustoichivosti i upravleniya, Nauka, Novosibirsk, 1991, 36–40

[10] A. S. Andreev, Ustoichivost neavtonomnykh funktsionalno-differentsialnykh uravnenii, Izd-vo UlGU, Ulyanovsk, 2005

[11] S. V. Pavlikov, “Predelnye uravneniya i funktsionaly Lyapunova v zadache ob ustoichivosti po chasti peremennykh”, Uch. zap. Ulyanovsk. gos. un-ta. Ser. Fundament. probl. matem. i mekh., 1:13 (2003), 63–74

[12] S. V. Pavlikov, Metod funktsionalov Lyapunova v zadachakh ustoichivosti, In-t upravleniya, Naberezhnye Chelny, 2006

[13] A. S. Andreev, “Metod funktsionalov Lyapunova v zadache ob ustoichivosti funktsionalno-differentsialnykh uravnenii”, Avtomat. i telemekh., 2009, no. 9, 4–55 | MR | Zbl

[14] O. Ignatyev, V. Madrekar, “Barbashin–Krasovskii theorem for stochastic differential equations”, Proc. of the Amer. Math. Soc., 138:11 (2010), 4123–4128 | DOI | MR | Zbl

[15] E. L. Panasenko, E. A. Tonkov, “Rasprostranenie teorem E. A. Barbashina i N. N. Krasovskogo ob ustoichivosti na upravlyaemye dinamicheskie sistemy”, Tr. IMM UrO RAN, 15, no. 3, 2009, 185–201

[16] G. Q. Xu, S. P. Yung, “Lyapunov stability of abstract nonlinear dynamic system in Banach space”, IMA J. Math. Control Inform., 20:1 (2003), 105–127 | DOI | MR | Zbl

[17] A. A. Ignatev, “Ob ekviasimptoticheskoi ustoichivosti po chasti peremennykh”, PMM, 63:5 (1999), 871–875 | MR | Zbl

[18] N. G. Bulgakov, B. S. Kalitin, “Obobschenie teorem vtorogo metoda Lyapunova. I. Teoriya”, Izv. AN BSSR. Ser. fiz.-matem. nauk, 1978, no. 3, 32–36 | MR | Zbl

[19] A. A. Kosov, “K metodu vektornykh funktsii Lyapunova”, Funktsii Lyapunova i ikh primenenie, Nauka, Novosibirsk, 1986, 106–110

[20] A. S. Andreev, S. V. Pavlikov, “Issledovanie ustoichivosti funktsionalno-differentsialnykh uravnenii na osnove znakopostoyannykh funktsionalov Lyapunova”, Tr. Srednevolzhsk. matem. obsch-va, 2:1 (1999), 74–75

[21] A. S. Andreev, S. V. Pavlikov, “Neznakoopredelennye funktsionaly Lyapunova v zadache ob ustoichivosti funktsionalno-differentsialnykh uravnenii s konechnym zapazdyvaniem”, MTT, 34 (2004), 112–120 | MR

[22] A. O. Ignatev, “Issledovanie ustoichivosti s pomoschyu znakopostoyannykh funktsii Lyapunova”, Ukr. matem. vestn., 2:1 (2005), 74–83 | MR | Zbl

[23] S. V. Pavlikov, “Znakopostoyannye funktsionaly Lyapunova v zadache ob ustoichivosti funktsionalno-differentsialnogo uravneniya”, PMM, 71:3 (2007), 377–388 | MR | Zbl

[24] A. O. Ignatyev, O. A. Ignatyev, “Stability investigation of invariant sets by means of semidefinite Lyapunov functions”, Nonlinear Anal. Real World Appl., 12:3 (2011), 1453–1458 | DOI | MR | Zbl

[25] J. Jiang, M. Han, “Small-amplitude limit cycles of some Liénard-type systems”, Nonlinear Anal., 71:12 (2009), 6373–6377 | DOI | MR | Zbl

[26] F. Dumortier, D. Panazzolo, R. Roussaie, “More limit cycles than expected in Liénard equations”, Proc. Amer. Math. Soc., 135:6 (2007), 1895–1904 | DOI | MR | Zbl

[27] G. A. Leonov, “Predelnye tsikly uravneniya Lenara s razryvnymi koeffitsientami”, DAN, 426:1 (2009), 47–50 | MR | Zbl

[28] J. Llibre, C. Valls, “On the local analytic integrability at the singular point of a class of Liénard analytic differential systems”, Proc. Amer. Math. Soc., 138:1 (2010), 253–261 | DOI | MR | Zbl

[29] A. Yu. Aleksandrov, “Ob ustoichivosti uravneniya Lenara s nestatsionarnymi vozmuscheniyami”, Differents. uravneniya, 32:5 (1996), 702–703 | MR | Zbl

[30] A. Yu. Aleksandrov, “Ob ustoichivosti vektornogo uravneniya Lenara s nestatsionarnymi vozmuscheniyami”, Sib. matem. zhurn., 40:5 (1999), 977–986 | MR | Zbl