Invariant Sets and Boundary Systems of Nonautonomous Differential Inclusions

TL;DR

This paper introduces a deterministic boundary system approach to analyze invariant sets in nonautonomous differential inclusions, linking them to normal cones and minimal attractors in dynamical systems.

Contribution

It establishes a novel finite-dimensional method connecting invariant sets of differential inclusions with boundary systems and normal cones, enhancing understanding of attractors.

Findings

Invariant sets lift to backward invariant normal cones in the boundary system.

Existence and uniqueness of minimal attractors are proven under exponential stability.

The unit normal bundle acts as a pullback attractor for the boundary system.

Abstract

In this paper we propose a finite-dimensional and deterministic approach to the study of invariant sets of certain nonautonomous differential inclusions naturally arising in the context of random and control dynamical systems, as well as in systems modeling the dynamical propagation of uncertainty. In particular, to any such differential inclusion, we associate a finite-dimensional and deterministic system of nonautonomous ordinary differential equations, which we call the boundary system, due to its following characteristic property: invariant sets of the differential inclusion lift in a unique way to backward invariant unit normal cones of the associated boundary system, and these are even invariant if the boundary is smooth. We further illustrate the strength of this approach in the study of minimal attractors of nonautonomous linear differential inclusions. Under the natural assumption of exponential stability for the unperturbed problem, we establish existence and uniqueness of a minimal attractor for the differential inclusion with fiberwise strictly convex, closed, and continuously differentiable boundaries. We also show that the unit normal bundle is in fact the pullback attractor for the skew-product flow associated to the boundary system which extends to the global attractor when the underlying admits a compact base.