Partial Asymptotic Stability

Abstract

A class of abstract dynamical systems with multivalued flows of solutions in a metric space is introduced in this chapter. For this class of systems, the property of partial asymptotic stability with respect to a continuous functional is studied. In order to characterize the limit set of a trajectory of a multivalued system, a modification of the invariance principle is proposed. This result is applied to derive sufficient conditions for partial asymptotic stability of an equilibrium by using a continuous Lyapunov functional. Such conditions are also formulated for particular classes of systems governed by differential inclusions, ordinary differential equations, and nonlinear semigroups in a Banach space. For further applications of these results to the partial stability analysis of nonlinear abstract differential equations, conditions for the relative compactness of trajectories are derived by considering nonlinear perturbations of dissipative operators. The partial stabilization problem is studied by using differentiable Lyapunov functions for control affine systems in a finite-dimensional space. This treatment is illustrated by examples of the attitude stabilization of a satellite controlled by thrust jets or flywheels.