Partial asymptotic stability and stabilization of nonlinear abstract differential equations

The problem of partial asymptotic stability with respect to a continuous functional is considered for a class of abstract multivalued systems on a metric space. Such a class includes nonlinear finite and infinite dimensional dynamical systems, differential inclusions, delay equations, etc. An extension of the invariance principle for multivalued dynamical systems is obtained provided that there is a continuous Lyapunov functional. By applying this technique, we derive a sufficient condition for partial asymptotic stability of the equilibrium in a metric space. For dynamical systems on a finite dimensional Euclidean space, the above result is analogous to the Risito-Rumyantsev theorem. The case of nonlinear continuous semigroups on a Banach space is characterized by means of differentiable Lyapunov functionals. Namely, if the above functional is positive definite with respect to a part of the state variables and its time-derivative is non-positive, then the semigroup is partially asymptotically stable under some extra assumptions on the semigroup trajectories. This result can be applied for solving the partial stabilization problem if strong stabilizability of a distributed system is not possible.