On the stability of functional-differential inclusions with the use of invariantly differentiable Lyapunov functionals. (English. Russian original) Zbl 1187.34084
Differ. Equ. 43, No. 8, 1079-1087 (2007); translation from Differ. Uravn. 43, No. 8, 1055-1063 (2007).
The functional differential inclusion with delay
\[ \dot{x}\in F(t,x_t(\cdot)),\quad x_{t_0}(\cdot)=\varphi_0(\cdot) \]
is considered. Here, \(x_t(\cdot)\in C_r, x_t(\theta)=x(t+\theta), -\tau\leq\theta\leq 0\), and \(\varphi(\cdot)\in C_r\) is the initial function. Problems of (weak) stability and (weak) asymptotic stability for such systems are studied. For autonomous functional differential inclusions with delay the principle of invariance is justified.
\[ \dot{x}\in F(t,x_t(\cdot)),\quad x_{t_0}(\cdot)=\varphi_0(\cdot) \]
is considered. Here, \(x_t(\cdot)\in C_r, x_t(\theta)=x(t+\theta), -\tau\leq\theta\leq 0\), and \(\varphi(\cdot)\in C_r\) is the initial function. Problems of (weak) stability and (weak) asymptotic stability for such systems are studied. For autonomous functional differential inclusions with delay the principle of invariance is justified.
Reviewer: Alexander O. Ignatyev (Donetsk)
MSC:
| 34K09 | Functional-differential inclusions |
| 34K20 | Stability theory of functional-differential equations |
References:
| [1] | Aubin, J.-P. and Ekeland, I., Applied Nonlinear Analysis, New York: John Wiley & Sons, 1984. Translated under the title Prikladnoi nelineinyi analiz, Moscow: Mir, 1988. · Zbl 0641.47066 |
| [2] | Kim, A.V., i-Gladkii analiz i funktsional’no-differentsial’nye uravneniya (i-Smooth Analysis and Functional-Differential Equations), Yekaterinburg: Russian Acad. of Sci., Ural Branch, Inst. Mat. and Mech., 1996. |
| [3] | Kurzhanskii, A.B., Differ. Uravn., 1970, vol. 6, no. 10, pp. 1800–1809. |
| [4] | Kolmanovskii, V.B. and Nosov, V.R., Ustoichivost’ i periodicheskie rezhimy reguliruemykh sistem s posledeistviem (Stability and Periodic Modes of Controlled Systems with Aftereffect), Moscow: Nauka, 1981. |
| [5] | Trubnikov, Yu.V. and Perov, A.I., Differentsial’nye uravneniya s monotonnymi nelineinostyami (Differential Equations with Monotone Nonlinearities), Minsk: Nauka i Tekhnika, 1986. · Zbl 0623.34002 |
| [6] | Filippov, A.F., Differentsial’nye uravneniya s razryvnoi pravoi chast’yu (Differential Equations with Discontinuous Right-Hand Sides), Moscow: Nauka, 1985. · Zbl 0571.34001 |
| [7] | Egorov, A.I., Obyknovennye differentsial’nye uravneniya (Ordinary Differential Equations), Moscow: Fizmatlit, 2005. |
| [8] | Rouche, N., Habets, P., and Laloy, M., Stability Theory by Ljapunov’s Direct Method, New York: Springer-Verlag, 1977. Translated under the title Pryamoi metod Lyapunova v teorii ustoichivosti, Moscow: Mir, 1980. · Zbl 0364.34022 |
| [9] | Hale, J., Theory of Functional Differential Equations, New York: Springer, 1977. Translated under the title Teoriya funktsional’no-differentsial’nykh uravnenii, Moscow: Mir, 1984. |
| [10] | Aizerman, M.A. and Pyatnitskii, E.S., Avtomat. i Telemekh., 1974, no. 7, pp. 33–47; no. 8, pp. 39–61. |
| [11] | Finogenko, I.A., Izv. Akad. Nauk Teor. Sist. Upr., 2004, no. 4, pp. 19–26. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
