Partial asymptotic stability of abstract differential equations. (Russian, English) Zbl 1126.34040

Ukr. Mat. Zh. 58, No. 5, 629-637 (2006); translation in Ukr. Math. J. 58, No. 5, 709-717 (2006).
Summary: We consider the problem of partial asymptotic stability with respect to a continuous functional for a class of abstract dynamical processes with multivalued solutions on a metric space. This class of processes includes finite- and infinite-dimensional dynamical systems, differential inclusions, and delay equations. We prove a generalization of the Barbashin-Krasovski theorem and the LaSalle invariance principle under the conditions of the existence of a continuous Lyapunov functional. In the case of the existence of a differentiable Lyapunov functional, we obtain sufficient conditions for the partial stability of continuous semigroups in a Banach space.


34G20 Nonlinear differential equations in abstract spaces
47E05 General theory of ordinary differential operators
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI


[1] Fattorini, H. O., Infinite-Dimensional Optimization and Control Theory (1999), Cambridge: Cambridge University Press, Cambridge · Zbl 0931.49001
[2] Luo, Z.-H.; Guo, B.-Z.; Morgul, O., Stability and Stabilization of Infinite-Dimensional Systems with Applications (1999), London: Springer, London · Zbl 0922.93001
[3] Rumyantsev, V. V.; Oziraner, A. S., Stability and Stabilization of Motion with Respect to a Part of Variables (1987), Moscow: Nauka, Moscow · Zbl 0626.70021
[4] Clarke, F. H.; Ledyaev, Yu. S.; Sontag, E. D.; Subbotin, A. I., Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Contr., 42, 1394-1407 (1997) · Zbl 0892.93053 · doi:10.1109/9.633828
[5] Shestakov, A. A., Generalized Direct Lyapunov Method for Systems with Distributed Parameters (1990), Moscow: Nauka, Moscow · Zbl 0688.93043
[6] LaSalle, J. P.; Cesari, L.; Hale, J. K.; LaSalle, J. P., Stability theory and invariance principles, Dynamical Systems, International Symposium on Dynamical Systems (Providence, 1974), 211-222 (1976), New York: Academic Press, New York · Zbl 0356.34047
[7] Hartman, P., Ordinary Differential Equations (1970), Moscow: Mir, Moscow · Zbl 0214.09101
[8] Barbashin, E. A.; Krasovskii, N. N., On the stability of motion on the whole, Dokl. Akad. Nauk SSSR, 86, 3, 453-456 (1952) · Zbl 0047.33001
[9] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), New York: Springer, New York · Zbl 0516.47023
[10] Ladyzhenskaya, O., Attractors for Semigroups and Evolution Equations (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0755.47049
[11] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Oxford (1981). · Zbl 0456.34002
[12] Barbu, V., Analysis and Control of Nonlinear Infinite-Dimensional Systems (1992), San Diego: Academic Press, San Diego
[13] Dafermos, C. M.; Slemrod, M., Asymptotic behavior of nonlinear contraction semi-groups, J. Funct. Anal., 13, 97-106 (1973) · Zbl 0267.34062 · doi:10.1016/0022-1236(73)90069-4
[14] A. L. Zuev, “Stabilization of nonautonomous systems with respect to a part of variables using controlled Lyapunov functions,” Probl. Uprav. Inform., No. 4, 25-34 (2000).
[15] Tolstonogov, A. A., Differential Inclusions in a Banach Space (1986), Novosibirsk: Nauka, Novosibirsk · Zbl 0689.34014
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.