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Geometric Hybrid Dynamical Systems: Part I - Modeling and Stability

Piyush P. Jirwankar, Daniel E. Ochoa, Ricardo G. Sanfelice

arXiv:2609.17687v2math.DScs.SYeess.SYmath.DG

TL;DR

Hybrid-system frameworks based on Euclidean embeddings or metric choices do not readily capture intrinsic dynamics on nonlinear, possibly disconnected manifolds. This paper develops a coordinate-independent hybrid-inclusion framework on C1-manifolds and establishes stability, Lyapunov, and invariance results within it.

  • Problem

    Existing approaches do not combine set-valued hybrid-inclusion generality with an intrinsic manifold treatment that accommodates disconnected state spaces, while metric choices can make stability analysis depend on noncanonical distances.

    Source: B. Existing Models of Hybrid Systems… · p. 1, A. On Embeddings and Metrizations · p. 1

  • Method

    The paper develops a coordinate-independent framework on C1-manifolds, with topological stability notions, nonsmooth Lyapunov analysis, and set-valued tools for solution and limit-set properties.

    Source: C. Contributions · p. 2, §VII · p. 15

  • Results

    The paper gives sufficient conditions for uniform global pre-asymptotic stability through a hybrid Lyapunov theorem and derives a hybrid invariance principle for geometric hybrid systems.

    Source: B. Hybrid Lyapunov Theorem on… · p. 11, §VI · p. 13

  • Takeaways & Limitations

    The framework separates the geometric structure needed for analysis from extrinsic choices such as a Riemannian metric or Euclidean embedding.

    Source: §VII · p. 15

Abstract

from arXiv · show

We present a framework for the modeling and analysis of geometric hybrid dynamical systems as hybrid inclusions on $C^1$-manifolds. Using tools from nonsmooth and set-valued analysis on manifolds, we derive coordinate-independent sufficient conditions for the existence of nontrivial solutions to this type of systems. We present geometric notions of uniform stability and attractivity of compact sets, and establish their equivalence to metric-based stability notions when the manifold is endowed with a Riemannian structure. We also introduce nonsmooth Lyapunov functions and a hybrid Lyapunov theorem providing sufficient conditions for uniform global asymptotic stability of compact sets. Finally, by characterizing $\omega$-limit sets of precompact solutions, we derive a hybrid invariance principle for geometric hybrid dynamical systems. The results are demonstrated through several running examples.

I. INTRODUCTION

The paper develops a coordinate-independent framework for hybrid dynamics on C1-manifolds, motivated by the limits of embedding-based and metric-dependent approaches. It combines set-valued modeling with geometric stability, Lyapunov, and invariance results without requiring an embedding or metric.

A. On Embeddings and Metrizations

B. Existing Models of Hybrid Systems on Manifolds

  • Earlier manifold hybrid models often use differential equations or semiflow relations, which are not naturally suited to nonunique solutions or perturbations.The paper situates its approach alongside embedded, fiber-bundle, topological-hybrid, and connected-Riemannian formulations.Source: B. Existing Models of Hybrid Systems… · p. 1

I. INTRODUCTION

  • The paper calls its coordinate-independent framework for hybrid systems evolving on C1-manifolds geometric hybrid dynamical systems.It is motivated by applications whose states evolve on nonlinear manifolds, where Euclidean frameworks do not readily transfer.Source: §I · p. 1, C. Contributions · p. 2

C. Contributions

  • The framework combines sufficient conditions for solution existence, topological stability notions, a hybrid Lyapunov theorem, and an invariance principle for systems on C1-manifolds.The authors also illustrate it with attitude stabilization on S3 and billiard stabilization on the Möbius band.Source: C. Contributions · p. 2
  • The contribution list also includes manifold nonsmooth-analysis tools and a closure property for solution sets, alongside two running examples.The examples concern hybrid attitude stabilization on S3 and a second-order billiard system with impacts on the Möbius band.Source: C. Contributions · p. 2

A. Notation

  • The notation section defines basic set and function terminology, including class-K functions, positive definiteness, properness, and set-valued-map domains and graphs.It also defines a compact neighborhood as a compact set whose interior contains the set of interest.Source: A. Notation · p. 2

B. Differential geometry

  • The differential-geometric preliminaries use the open Möbius band as a running manifold example, constructed by identifying opposite ends with the transverse coordinate reversed.The paper describes the band as a quotient space equipped with smooth charts.Source: B. Differential geometry · p. 3, Figure 1 · p. 3

C. Set-valued analysis

  • The set-valued analysis preliminaries define convergence through limits of graphs and establish manifold versions of set-valued-analysis tools.They introduce outer semicontinuity and relate graphical convergence to convergence of domains and ranges.Source: C. Set-valued analysis · pp. 3–4

III. GEOMETRIC HYBRID DYNAMICAL SYSTEMS

The section models geometric hybrid dynamical systems on finite-dimensional C1-manifolds using flow and jump data, and defines their solutions on hybrid time domains. It gives regularity conditions and illustrates the framework with quaternion stabilization and billiard motion on a Möbius band.

System model

  • A geometric hybrid system H=(C,F,D,G,M) has flow and jump sets, a set-valued flow map into each state’s tangent space, and a set-valued jump map on a finite-dimensional C1-manifold.The flow map satisfies F(x) ⊂ T_xM.Source: §III · p. 4

Solutions

  • Solutions are hybrid arcs that evolve locally absolutely continuously during flow and obey the system’s flow and jump rules on hybrid time domains indexed by time and jump count.The section defines total flow time, number of jumps, solution maximality, completeness, and precompactness.Source: §III · pp. 4–5

Basic conditions

  • The geometric hybrid basic conditions require a C1-manifold, relatively closed flow and jump sets, and outer semicontinuous, locally precompact maps with convex flow values.In this manifold setting, a distance-dependent notion of well-posedness is not defined; a weaker metric-independent notion based on sequential closedness of compact solutions is cited.Source: §III · p. 5

Quaternion stabilization

  • The quaternion stabilization example uses the state space S3 × {−1,1} and a logic-based hybrid controller to form a geometric hybrid system.Its closed-loop flow and jump data satisfy the geometric hybrid basic conditions.Source: §III · p. 5

Billiard on the Möbius band

  • The Möbius-band billiard model evolves position and velocity on a tangent bundle, with impacts and mode changes represented by distinct jump triggers.At impacts, the reset shifts position inward, scales the velocity component perpendicular to the boundary by an uncertain restitution coefficient, and parallel-transports the result to the new tangent space.Source: §III · pp. 5–6

IV. BASIC PROPERTIES OF THE SOLUTION SET OF H

The section establishes basic properties of the solution set, including sufficient conditions for nontrivial solutions on a manifold and a classification of maximal solutions. Its existence argument transports a tangent-cone viability test through local coordinates.

IV. BASIC PROPERTIES OF THE SOLUTION SET OF H

  • The solution set is shown to be closed by a sequential argument under the geometric hybrid basic conditions.The section also states that the manifold tangent cone is defined independently of the chosen coordinate chart.Source: §IV · p. 6, A. Existence of Solutions to H · p. 6

A. Existence of Solutions to H

  • A nontrivial solution exists from a flow-set point when the flow map intersects the manifold tangent cone locally; the condition is checked in coordinates and transferred back to the manifold.If the condition holds at every point of C\D, solutions exist from every point of C∪D.Source: A. Existence of Solutions to H · p. 6
  • Each maximal solution is complete, has a bounded domain whose final flow arc is maximal, or ends at a finite hybrid time outside C∪D.The latter two alternatives are ruled out, respectively, if C is compact or G(D)⊂C∪D.Source: A. Existence of Solutions to H · pp. 6–7

B. Sequential Closedness of the Solution Set of H

The section characterizes graphical limits of solution sequences under the geometric hybrid basic conditions. With local eventual precompactness, the limit is a solution; without it, the sequence exhibits an escape-to-horizon alternative.

B. Sequential Closedness of the Solution Set of H

  • When a graphically convergent solution sequence is locally eventually precompact and does not escape to the horizon, its graphical limit is a solution to H.Local eventual precompactness requires the tails of the sequence to remain in a common compact set on each bounded hybrid-time window.Source: B. Sequential Closedness of the… · p. 9
  • Without local eventual precompactness, a sequence of solutions admits times approaching a finite horizon, and the limit restricted before that horizon is a maximal solution.For each compact K, the limit arc eventually avoids K near the horizon.Source: B. Sequential Closedness of the… · p. 9
  • The graphical-limit result is identified as nominal well-posedness of H.The passage relates this property to topological consequences for the solution-set map.Source: B. Sequential Closedness of the… · p. 9

V. LYAPUNOV THEOREM FOR GEOMETRIC HYBRID DYNAMICAL SYSTEMS

The section introduces topological stability and attractivity notions to formulate a hybrid Lyapunov theorem for uniform global asymptotic stability of compact sets.

V. LYAPUNOV THEOREM FOR GEOMETRIC HYBRID DYNAMICAL SYSTEMS

  • The proposed stability and attractivity notions are topological and independent of a metric on the manifold.They are introduced to support a hybrid Lyapunov theorem certifying uniform global asymptotic stability of a compact set.Source: §V · p. 9
  • The section defines uniform stability, attractivity, and asymptotic stability for compact sets.Source: §V · p. 9
  • The stated objective is a hybrid Lyapunov theorem providing sufficient conditions for uniform global asymptotic stability.Source: §V · p. 9

A. Stability and Attractivity Notions on C1-Manifolds

The section defines stability and attractivity using neighborhoods on C1-manifolds, allowing these notions to be stated without a distance function. It distinguishes pre-asymptotic notions, which accommodate maximal solutions that may be incomplete.

A. Stability and Attractivity Notions on C1-Manifolds

  • Uniform global stability combines uniform Lyapunov stability, which keeps solutions from W inside U, with uniform Lagrange stability, which bounds solutions from compact W in compact U.On complete Riemannian manifolds with compact A, the definition is equivalent to a class-K∞ bound on Riemannian distance at each hybrid time.Source: A. Stability and Attractivity Notions… · p. 9
  • Pre-attractivity requires precompact ranges for finite-length solutions and eventual entry into every neighborhood of A for complete solutions starting nearby.Source: A. Stability and Attractivity Notions… · p. 9
  • Uniform global pre-attractivity requires that solutions from any compact neighborhood of A enter each prescribed compact neighborhood after a uniform hybrid-time bound.Source: A. Stability and Attractivity Notions… · p. 9
  • Pre-asymptotic stability combines uniform Lyapunov stability and pre-attractivity, while uniform global pre-asymptotic stability combines uniform global stability and uniform global pre-attractivity.The prefix “pre-” allows maximal solutions to be incomplete; it is dropped when all maximal solutions are complete.Source: A. Stability and Attractivity Notions… · pp. 9–10

B. Hybrid Lyapunov Theorem on C1-Manifolds

The section develops nonsmooth Lyapunov tools on C1-manifolds and gives sufficient conditions for uniform global pre-asymptotic stability of a compact set. The theorem allows several combinations of flow and jump behavior, provided the applicable decrease or timing conditions hold.

Hybrid Lyapunov Theorem

  • A proper Lyapunov candidate certifies uniform global pre-asymptotic stability when any one of the theorem’s decrease and timing conditions is satisfied.The candidates may be locally Lipschitz, rather than continuously differentiable.Source: B. Hybrid Lyapunov Theorem on… · pp. 10–11
  • Strict decrease during both flows and jumps is one sufficient condition for stability.The theorem also provides alternatives that allow no increase in one phase, balanced growth across phases, or bounds on phase duration or jump count.Source: B. Hybrid Lyapunov Theorem on… · p. 11
  • Strict flow decrease with no jump increase suffices when solutions satisfy a lower bound on elapsed flow time relative to hybrid time.For each compact neighborhood, the condition uses a class-K-infinity function and a nonnegative constant.Source: B. Hybrid Lyapunov Theorem on… · p. 11
  • Strict jump decrease with no flow increase suffices when solutions satisfy a lower bound on jump count relative to hybrid time.The condition applies for each compact neighborhood of the target set.Source: B. Hybrid Lyapunov Theorem on… · p. 11
  • Flow and jump growth may be balanced if their weighted cumulative rates decay along every solution.The bounds constrain flow and jump changes separately, while a further inequality requires their combined contribution to decrease with hybrid time.Source: B. Hybrid Lyapunov Theorem on… · p. 11
  • Strict decrease in one phase can also suffice when the other phase is bounded in duration or jump count.The theorem gives a bounded-flow-time option with strict jump decrease and a finite-jump option with strict flow decrease.Source: B. Hybrid Lyapunov Theorem on… · p. 11

VI. HYBRID INVARIANCE PRINCIPLE

The section develops an invariance principle for geometric hybrid systems to address Lyapunov functions that are only nonincreasing along flows and jumps. It characterizes omega-limit sets and shows, under stated assumptions, how they relate to weakly invariant sets; a Möbius-band billiard illustrates the result.

VI. HYBRID INVARIANCE PRINCIPLE

  • The principle addresses cases where Lyapunov decrease conditions are only nonstrict, extending an invariance result previously available for Euclidean hybrid systems to the manifold setting.The section first establishes properties of omega-limit sets for complete, precompact solutions satisfying the geometric hybrid basic conditions.Source: §VI · p. 13
  • For complete, precompact solutions, the omega-limit set is nonempty, compact, weakly invariant, and eventually approached by the solution.Every open neighborhood of the set contains all sufficiently late solution points.Source: §VI · p. 13
  • When a continuous function is nonincreasing along a hybrid arc, its omega-limit set lies in a single level set of that function.The section uses this characterization as a step toward the hybrid invariance principle.Source: §VI · p. 13
  • For a complete, precompact solution in U with nonpositive flow and jump changes of V, Theorem 6.5 identifies its omega-limit set as the largest weakly invariant set in the theorem’s specified set.The theorem assumes the geometric hybrid basic conditions; its conclusion concerns a level r in the range of V on U ∩ (C ∪ D ∪ G(D)).Source: §VI · pp. 13–14
  • In the Möbius-band billiard example, the constructed Lyapunov function has strictly negative changes at jumps, and every maximal solution is complete and precompact.The omega-limit set reduces to X, defined by a level set of V intersected with the set where its flow derivative vanishes; the example certifies A as asymptotically stable.Source: §VI · pp. 14–15

VII. CONCLUSION

The paper proposes a framework for modeling and analyzing hybrid systems whose states evolve on C^1-manifolds. It separates the minimal geometric structure needed for analysis from extrinsic structure, and identifies robustness and input extensions as future work.

VII. CONCLUSION

  • The framework models and analyzes hybrid systems evolving on C^1-manifolds while separating minimal geometric structure from assumptions not intrinsic to the dynamics.It is independent of extrinsic structure such as a Riemannian metric or an embedding into Euclidean space.Source: §VII · p. 15
  • The conclusion presents the framework as a geometric approach whose analysis does not require an embedding into Euclidean space.It also states that the framework does not depend on a Riemannian metric.Source: §VII · p. 15
  • Future work includes formulating robustness properties for structural perturbations and extending the framework to systems with inputs.The proposed input extension would enable control-synthesis methods and input-to-state stability certificates for measurable disturbances.Source: §VII · p. 15

APPENDIX A PROOF OF THEOREM 4.6

Appendix A proves Theorem 4.6 by extending graphical-limit results for constrained differential inclusions and then accounting for hybrid jumps. Its intermediate results handle sequences with common or varying time domains and cases involving escape to the horizon.

APPENDIX A PROOF OF THEOREM 4.6

  • The proof of Theorem 4.6 combines graphical-limit results for continuous-time constrained differential inclusions with an additional step accounting for jumps.The appendix first treats solutions with the same time domain, then varying time domains, and then sequences that are not locally eventually precompact and do not escape to the horizon.Source: Appendix A · p. 16
  • For continuous-time systems with empty jump set, a graphical limit of locally eventually precompact solution sequences with a common compact interval as domain is a solution.Proposition A.2 extends the result to graphically convergent, locally eventually precompact sequences with different time domains.Source: Appendix A · pp. 16–17
  • For a graphically convergent sequence of solutions that is not locally eventually precompact and does not escape to the horizon, Proposition A.3 establishes a smallest time t*>0 at which escape occurs.Source: Appendix A · p. 17

APPENDIX B AUXILIARY RESULTS ON GEOMETRIC NOTIONS

Appendix B supplies auxiliary geometric results, including coordinate independence of a tangent-cone construction and a manifold-wide Rademacher theorem. Under stated Riemannian assumptions, it also establishes equivalence between topological and metric characterizations of uniform global stability.

APPENDIX B AUXILIARY RESULTS ON GEOMETRIC NOTIONS

  • The tangent-cone construction in Definition 4.1 is coordinate-independent: either chart at a point yields the same tangent space representation.The proof establishes this by transporting the tangent cone through the differential of the chart transition map.Source: Appendix B · p. 19
  • A locally Lipschitz function between C^1-manifolds is differentiable almost everywhere on the domain manifold.The appendix notes that its proof gives a manifold-wide conclusion, complementing a result proved within a single chart.Source: Appendix B · p. 19
  • On a complete Riemannian manifold satisfying Assumption B.3, a compact set is UGS if and only if a class-K∞ bound relates solution distance from the set to initial distance.Assumption B.3 requires finitely many connected components and that the distance restrict to the Riemannian distance on each component.Source: Appendix B · p. 19

APPENDIX C HYBRID CONTROLLER FOR QUATERNION STABILIZATION

The appendix gives the hybrid controller used for quaternion stabilization in Example 3.5 and establishes continuity of the map Φ in Example 4.4. It also collects auxiliary results, including criteria for escape to the horizon and graphical subsequence convergence.

IN EXAMPLE 3.5

  • In Example 3.5, the hybrid controller has state h, input q, output ξ, flow dynamics ḣ = 0, jump dynamics h⁺ = −h, and output ξ = −hKεε.Kε is symmetric positive definite, and interconnection with quaternion kinematics gives the system described in Example 3.5.Source: IN EXAMPLE 3.5 · p. 20

APPENDIX D EXAMPLE 4.4 CONTINUED: CONTINUITY OF Φ

  • In Example 4.4, continuity of Φ follows by showing that parallel transport P depends continuously on its initial tangent vector.The argument uses an ODE whose coefficients are smooth because the Levi-Civita Christoffel symbols are smooth.Source: Appendix D · p. 20

APPENDIX E COMPENDIUM OF AUXILIARY RESULTS

  • Appendix E compiles auxiliary results on set-valued analysis, graphical convergence, comparison functions, Hopf–Rinow, and geodesic convexity.These results are presented as tools for the paper’s analysis on Riemannian manifolds.Source: Appendix E · p. 20

1. Auxilliary results on graphical convergence

  • A sequence of sets escapes to the horizon if and only if its limit is empty.The proof connects failure to escape with a nonempty upper limit, and conversely uses a precompact neighborhood to rule out a nonempty limit.Source: §1 · p. 20
  • Hybrid arcs with initial points in a common compact set have a graphically convergent subsequence.The proof invokes the graphical-convergence result for sequences that do not escape to the horizon.Source: §1 · pp. 20–21

2. Auxiliary Results on Comparison Functions

The auxiliary results bound possibly discontinuous comparison functions by class K∞ functions away from zero, characterize metric inflations of compact sets, and provide local geodesic-convex neighborhoods with coordinate bounds.

2. Auxiliary Results on Comparison Functions

  • For any r > 0, a nondecreasing, possibly discontinuous function γ that diverges at infinity admits an upper bound α in K∞ for all s ≥ r.The lemma’s bound applies beyond r, rather than requiring γ itself to be continuous.Source: §2 · p. 21

3. Metric Inflations of Compact Subsets of Riemannian Manifolds

  • For a nonempty compact set A in the stated complete Riemannian setting, each positive-radius closed metric inflation is a compact neighborhood of A.Every neighborhood of A contains some such inflation.Source: §3 · p. 21

4. Geodesic Convexity

  • Every point in an open set has a geodesically convex neighborhood contained in that set and in its connected component.The local geodesic-convexity result is stated for the Riemannian setting satisfying Assumption B.3.Source: §4 · p. 22
  • Near each point, a geodesically convex subset can be chosen in a chart with positive constants c and C, where c ≤ C.The lemma supplies these constants in its local coordinate characterization; the proof uses straight coordinate segments and minimizing geodesics.Source: §4 · p. 22

APPENDIX F EQUIVALENT CHARACTERIZATIONS

Appendix F gives coordinate-based characterizations of coordinate-independent notions. For local precompactness, it establishes equivalence with local boundedness of the map in suitable coordinates.

1. Local Precompactness

  • A set-valued map F from an n-dimensional C1 manifold into its tangent bundle is locally precompact exactly when its coordinate representation is locally bounded in suitable charts.The coordinate bound is by a constant multiple of the unit ball in R^n.Source: §1 · p. 23
  • A regular coordinate ball can be chosen around any point inside any open neighborhood, supporting the local chart construction.The tangent-bundle trivialization defined by a chart is a homeomorphism to the chart domain times R^n.Source: §1 · p. 23
  • The local-precompactness characterization uses a regular coordinate ball around each point and a local trivialization of the tangent bundle.The trivialization maps tangent vectors to their coordinate components, allowing compactness to be related to bounds in R^n.Source: §1 · p. 23

2. Local Absolute Continuity

The section characterizes local absolute continuity of manifold-valued functions through their coordinate representations. It also establishes that Ck functions on a manifold have Ck coordinate representations.

2. Local Absolute Continuity

  • For a Ck function f on a manifold, its representation f ◦ ψ⁻¹ is Ck in any chart (V, ψ) at the point.The section verifies this by comparing the chosen chart with charts witnessing the manifold definition of Ck regularity.Source: §2 · p. 24
  • A function into a Ck manifold is locally absolutely continuous exactly when its coordinate representation is locally absolutely continuous on each connected component of the preimage of a suitable chart.The characterization applies at every point in the function’s range.Source: §2 · p. 24
  • The converse direction obtains local absolute continuity by covering compact subintervals with chart preimages, partitioning them, and transferring absolute continuity through coordinate changes.The argument uses a finite subcover and a Lebesgue-number partition to reduce the proof to chartwise intervals.Source: §2 · pp. 24–25

3. Locally Lipschitz Functions

The section relates coordinate-based local Lipschitzness to Riemannian distance inequalities, then characterizes proper positive-definite functions on complete Riemannian manifolds through class-K∞ lower bounds.

3. Locally Lipschitz Functions

  • On Riemannian manifolds satisfying Assumption B.3, coordinate-defined local Lipschitzness is equivalent to a local distance inequality.The result generalizes [47, Prop. B.4], which assumes connected manifolds and establishes only one implication.Source: §3 · p. 26

4. Riemannian Equivalence of Proper Functions

  • For a continuous V in PD(A) on a complete Riemannian manifold satisfying Assumption B.3, properness is equivalent to an α in K∞ lower bound in distance to A.The bound is α(|x|_A) ≤ V(x) for every x in M.Source: §4 · pp. 26–27
  • An α lower bound makes each sublevel set bounded; continuity makes it closed, and completeness yields compactness.The proof places the sublevel set inside a compact distance neighborhood of A.Source: §4 · p. 27
  • Conversely, properness supports constructing a class-K∞ lower bound from distance to A.The proof builds a positive-definite, radially unbounded lower bound and then a K∞ function beneath it.Source: §4 · p. 27

APPENDIX G EXAMPLE 6.6 CONTINUED: ˙V COMPUTATION

The computation evaluates the derivative of V along the specified vector field. It uses geodesic motion to eliminate the kinetic-energy contribution and bounds the remaining term.

APPENDIX G EXAMPLE 6.6 CONTINUED: V̇ COMPUTATION

  • The computation establishes that V̇(x) ≤ 0 for each x ∈ C, expressing the derivative as dV_y(S(y)) plus dV_y(vlift(−grad V_q(y_1)−b y_2)).Source: Appendix G · p. 27
  • The geodesic spray makes the kinetic-energy derivative vanish, while the remaining term uses v = −grad Vq(y1) − by2 and metric symmetry.The computation identifies dK_y(S(y)) = 0 and simplifies the vertical-lift contribution.Source: Appendix G · pp. 27–28
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