Partial asymptotic stability of abstract differential equations
- 47 Downloads
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-Krasovskii 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.
KeywordsCauchy Problem Singular Point Classical Solution Asymptotic Stability Differential Inclusion
Unable to display preview. Download preview PDF.
- 6.J. P. LaSalle, “Stability theory and invariance principles,” in: L. Cesari, J. K. Hale, and J. P. LaSalle (editors), Dynamical Systems, International Symposium on Dynamical Systems (Providence, 1974), Vol. 1, Academic Press, New York (1976), pp. 211–222.Google Scholar
- 11.V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Oxford (1981).Google Scholar
- 12.V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, San Diego (1992).Google Scholar
- 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).Google Scholar