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Nagumo-Type Characterization of Forward Invariance for Constrained Systems

Olayo Reynaud, Mohamed Maghenem, Adnane Saoud, Sadek Belamfedel Alaoui, Ahmad Hably

TL;DR

This paper extends Nagumo's forward-invariance framework to constrained differential inclusions by proposing a Nagumo-type condition F(x) ⊂ T_K(x) on ∂K ∩ int(C). While this condition is generally only necessary, the authors introduce propagation, non-tangentiality, and transversality assumptions that relate the constrained system to the geometry of the intersection K ∩ C, making the tangential condition also sufficient. They develop a structured argument based on critical boundary points P = ∂K ∩ ∂C and the velocity set F_φ, supported by auxiliary lemmas on tangent/constrained cones, to prove the forward-invariance equivalence under these assumptions. The results provide a principled method to certify invariance in constrained settings, with implications for applications in networks, robotics, and control where state constraints are present.

Abstract

This paper proposes a Nagumo-type invariance condition for differential inclusions defined on closed constraint sets. More specifically, given a closed set to render forward invariant, the proposed condition restricts the system's dynamics, assumed to be locally Lipschitz, on the boundary of the set restricted to the interior of the constraint set. In particular, when the boundary of the set is entirely within the interior of the constraint set, the proposed condition reduces to the well-known Nagumo condition, known to be necessary and sufficient for forward invariance in this case. This being said, the proposed condition is only necessary in the general setting. As a result, we provide a set of additional assumptions relating the constrained system to the set to render forward invariant, and restricting to the geometry at the intersection between the two sets, so that the equivalence holds. The importance of the proposed assumptions is illustrated via examples.

Nagumo-Type Characterization of Forward Invariance for Constrained Systems

TL;DR

This paper extends Nagumo's forward-invariance framework to constrained differential inclusions by proposing a Nagumo-type condition F(x) ⊂ T_K(x) on ∂K ∩ int(C). While this condition is generally only necessary, the authors introduce propagation, non-tangentiality, and transversality assumptions that relate the constrained system to the geometry of the intersection K ∩ C, making the tangential condition also sufficient. They develop a structured argument based on critical boundary points P = ∂K ∩ ∂C and the velocity set F_φ, supported by auxiliary lemmas on tangent/constrained cones, to prove the forward-invariance equivalence under these assumptions. The results provide a principled method to certify invariance in constrained settings, with implications for applications in networks, robotics, and control where state constraints are present.

Abstract

This paper proposes a Nagumo-type invariance condition for differential inclusions defined on closed constraint sets. More specifically, given a closed set to render forward invariant, the proposed condition restricts the system's dynamics, assumed to be locally Lipschitz, on the boundary of the set restricted to the interior of the constraint set. In particular, when the boundary of the set is entirely within the interior of the constraint set, the proposed condition reduces to the well-known Nagumo condition, known to be necessary and sufficient for forward invariance in this case. This being said, the proposed condition is only necessary in the general setting. As a result, we provide a set of additional assumptions relating the constrained system to the set to render forward invariant, and restricting to the geometry at the intersection between the two sets, so that the equivalence holds. The importance of the proposed assumptions is illustrated via examples.

Paper Structure

This paper contains 14 sections, 6 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Rationale
  5. Assumptions
  6. Main Result
  7. Proofs
  8. Proof of Theorem \ref{['theo_main_result']}
  9. Conclusion
  10. Acknowledgements
  11. Intersection of contingent cones
  12. Proof of Proposition \ref{['propAp']}
  13. Proof of Proposition \ref{['lem_star_implies_not_leaving']}
  14. Proof of Proposition \ref{['lem_clFphi_supersets']}

Key Result

Proposition 1

Consider closed subsets $K$, $C \subset \mathbb{R}^n$, let $\mathcal{P} \subset \partial K \cap \partial C$ such that (Pr) holds, and let $F : \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ be continuous such that eqTangCond_motivation holds. Then, for all $x \in \mathcal{P}$, $\square$

Figures (1)

  • Figure 1: Phase portrait of the system in Example \ref{['example_no_transversality']}. Assumption \ref{['ass_transversality']} is not verified because $C$ and $\partial K$ intersect tangentially.

Theorems & Definitions (16)

  • Definition 1: Concept of solution
  • Definition 2: Forward invariance
  • Remark 1
  • Remark 2
  • Proposition 1
  • Theorem 1
  • Example 1
  • Example 2
  • Proposition 2
  • Proposition 3
  • ...and 6 more