Invariant Sets and Boundary Systems of Nonautonomous Differential Inclusions

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.

• The paper introduces a deterministic boundary system that links invariant sets of nonautonomous differential inclusions with a normalized Pontryagin Maximum Principle framework.
• It establishes a correspondence between set-valued dynamics and classical ODE flows, demonstrating invariant attractors with strictly convex, C1-smooth boundaries.
• The approach enables efficient computation and rigorous analysis of boundary bifurcations and stability in systems subject to uncertainty and control.

Invariant Sets and Boundary Systems of Nonautonomous Differential Inclusions

Motivation and Problem Statement

The paper "Invariant Sets and Boundary Systems of Nonautonomous Differential Inclusions" (2604.26713) addresses the structure and analysis of invariant sets for nonautonomous differential inclusions of the form

$\dot{x} \in \overline{B_\rho}(f(t, x)),$

in finite-dimensional spaces, where the perturbation (noise, control, or uncertainty) is uniformly bounded. These systems emerge naturally in random and control dynamical systems, as well as models of uncertainty propagation. A central objective is to rigorously connect the differentiable structure of their invariant sets, particularly the boundaries, to a deterministic, finite-dimensional ODE system—the so-called "boundary system."

Traditional approaches to analyzing invariant sets in set-valued dynamics rely on topological methods applied to infinite-dimensional hyperspaces. However, this framework restricts access to the finer geometric and differentiable features of invariant sets, such as boundary regularity, stability, and smooth bifurcations. The present work circumvents these barriers by constructing a deterministic boundary flow on the unit tangent bundle, enabling a novel, tractable avenue for both analysis and computation.

The Boundary System: Construction and Properties

For the given nonautonomous differential inclusion, the authors construct an associated boundary system—a nonautonomous system of ODEs on $$T_1 \mathbb{R}^d = \mathbb{R}^d \times \mathbb{S}^{d-1}$$:

$$\begin{aligned} \dot{x} &= f(t, x) + \rho n, \ \dot{n} &= -D_x f(t, x)^\top n + \langle n, D_x f(t, x)^\top n \rangle n, \end{aligned}$$

where $n$ denotes the outward unit normal vector. This system is a normalized form of the Pontryagin Maximum Principle (PMP) Hamiltonian flow associated to an optimal control problem minimizing the distance to the boundary of a reachable set.

The main result is a correspondence principle: invariant compact (nonautonomous) sets for the inclusion yield uniquely defined, possibly time-dependent, backward-invariant normal bundles for the boundary system; if the boundary is $C^1$, invariance is also forward. The technical backbone relies on the PMP and the geometry of Mordukhovich normal cones, accommodating nonsmooth boundaries.

This formulation allows for the propagation and stability analysis of differentiable structures on invariant set boundaries by examining single-valued deterministic ODE flows rather than set-valued dynamics in hyperspace. Notably, the boundary system is twice the phase-space dimension, yet is amenable to classical ODE tools.

Minimal Attractors in Linear Nonautonomous Systems

The theory is exemplified and strengthened in the context of nonautonomous linear inclusions ($f(t,x) = L(t)x$). Assume exponential stability of the unperturbed system:

$$\dot{x} = L(t)x, \quad \| \Psi(t, s) \| \leq K e^{-\gamma (t-s)}.$$

The paper proves:

• Existence, uniqueness, and explicit formulae for a minimal pullback attractor bundle $\{A_t\}$, each strictly convex and closed, given by

$$A_t = \left\{ x \mid x = \rho \int_{-\infty}^t \Psi(t,s)\xi(s) ds, \ \xi \in U \right\},$$

where $U$ enforces $$T_1 \mathbb{R}^d = \mathbb{R}^d \times \mathbb{S}^{d-1}$$0.

• The boundary of each fiber $$T_1 \mathbb{R}^d = \mathbb{R}^d \times \mathbb{S}^{d-1}$$1 is $$T_1 \mathbb{R}^d = \mathbb{R}^d \times \mathbb{S}^{d-1}$$2-smooth and strictly convex, with uniqueness of normal directions.
• The boundary system admits a well-defined deterministic flow whose global attractor is exactly the outward unit normal bundle of the attractor boundary.
• Forward and pullback convergence of reachable sets to the attractor, with convergence rates induced by exponential dichotomy.

Notably, the result extends classical regularity theory, as the associated Hamilton-Jacobi-Bellman PDE for the support function has a classical (not merely viscosity) solution on the boundary, due to strict convexity and control regularity.

Theoretical and Computational Implications

The construction translates set-valued (and possibly non-smooth) invariant set evolution into the analysis of a deterministic, finite-dimensional ODE system governing boundary and normal bundle dynamics. This perspective has several important implications:

• Regularity and Bifurcation Analysis: Access to the differentiable structure provides tools to study smooth stability, boundary singularities, and parametrized (smooth/topological) bifurcations of invariant sets—problems intractable in the hyperspace framework.
• Efficient Computation: Boundary evolution and attractor numerics can be computed via integrating the boundary system, as opposed to expensive inner-approximation methods required for set-valued flows.
• Pullback and Global Attractors: For linear nonautonomous inclusions, the boundary system's unit normal bundle serves as the pullback and global attractor for the associated skew-product flow, enabling rigorous approximation strategies and insights into long-time boundary behavior.

Extensions and Future Developments

The methodology directly generalizes to autonomous inclusions and is anticipated to extend to inclusions with non-spherical or even non-convex control sets, albeit with a modified boundary system reflecting the support function geometry.

The formalism opens the way for systematic study of bifurcations (especially boundary and topological transitions) of minimal invariant sets in time-dependent, uncertain, or controlled settings—key for understanding nonautonomous tipping points and robust control/viability analyses. Additionally, the geometric framework connects with contact and Legendrian theory in the unit tangent bundle, promising further advances in singularity theory of invariant sets.

Conclusion

The paper establishes a deterministic, finite-dimensional boundary system for nonautonomous differential inclusions, firmly linking the differentiable features of invariant set boundaries to the flow of a boundary ODE system based on the Pontryagin Maximum Principle. In the linear exponentially stable case, attractors have strictly convex, $$T_1 \mathbb{R}^d = \mathbb{R}^d \times \mathbb{S}^{d-1}$$3 boundaries and the outward normal bundle is a global attractor for the corresponding boundary flow.

This framework bridges a gap between set-valued and differentiable dynamical systems theory, enabling both deeper qualitative analysis and tractable computation of invariant sets under nonautonomous, controlled, or uncertain evolution. The theory has direct technical relevance for control, viability, and random dynamical systems, and sets the stage for new developments in the differentiable bifurcation theory of invariant sets.