Invariant sets and boundary systems of nonautonomous differential inclusions
File(s)
Accepted version
Author(s)
Kourliouros, Konstantinos
Longo, Iacopo P
Rasmussen, Martin
Type
Journal Article
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 as sociated to the boundary system which extends to the global attractor when
the underlying admits a compact base.
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 as sociated to the boundary system which extends to the global attractor when
the underlying admits a compact base.
Date Issued
2026-04-29
Date Acceptance
2026-09-21
Citation
Journal of Differential Equations
ISSN
0022-0396
Publisher
Elsevier
Journal / Book Title
Journal of Differential Equations
Copyright Statement
Copyright This paper is embargoed until publication. Once published the Version of Record (VoR) will be available on immediate open access.
License URL
Subjects
2020 Mathematics Subject Classification. 34A60 37C60 37C70 37N35 Nonautonomous dynamical systems
differential inclusions
invariant sets
attractors
boundary differential equation
set-valued dynamical systems
bounded noise
bounded controls
Pontryagin Maximum Principle
Publication Status
Accepted