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

Zuev, A. L.

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.

MSC:

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

Keywords:

asymptotic stability; Lyapunov functional; Banach space; differential equation

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