Set-Valued Lie Brackets: Generalizations & Applications

• Set-valued Lie brackets are generalized operations that output a convex set of directional limits, redefining classical commutators under nonsmooth and noncommutative settings.
• They play a crucial role in control theory by enabling degree-k control Lyapunov functions and ensuring global asymptotic controllability and Hölder regularity in PDEs.
• Emerging from nonsmooth analysis, quantum algebra, and deformation theory, these brackets provide a structured replacement for single-valued Lie brackets, preserving geometric and algebraic information.

Searching arXiv for recent and foundational papers on set-valued Lie brackets and closely related generalizations. Set-valued Lie brackets are generalized bracket operations in which a pair, or more generally an iterated tuple, of vector fields or generalized vector fields is assigned a subset of admissible bracket values rather than a single vector. In the literature represented here, this phenomenon has at least three distinct sources: nonsmoothness of vector fields, where classical derivatives exist only almost everywhere; noncommutative antisymmetrization, where closure depends on a noncanonical projector; and deformation theory, where infinitesimal perturbations of brackets are controlled only up to homotopy. Across these settings, set-valuedness is not treated as a purely pathological loss of uniqueness, but as a structured replacement for the classical Lie bracket that preserves geometric or algebraic information needed for controllability, regularity, or cohomological analysis (Bardi et al., 2019, Motta et al., 2016, Flood et al., 2023, Abad et al., 2010).

1. Nonsmooth definition and basic construction

For $C^1$ vector fields $f_1,f_2$, the classical Lie bracket is

$[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$

When the fields are only locally Lipschitz, the derivatives entering this formula need not exist everywhere. The set-valued replacement used in nonsmooth control and PDE theory is obtained by taking limits of classical brackets along nearby differentiability points and then convexifying: $$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$ Here $\mathrm{Diff}(f)$ denotes the set of differentiability points of $f$, and $\operatorname{co}$ is the convex hull. This construction appears explicitly in the analysis of degree-$k$ control Lyapunov functions for locally Lipschitz systems and in the regularity theory of degenerate eikonal equations (Motta et al., 2016, Bardi et al., 2019).

The same approach extends to higher iterated brackets. For example, for a third-order expression of the form $[ [f_1,f_2],f_3]$, with $f_1,f_2\in C^{1,1}$ and $f_1,f_2$0, the set-valued bracket is defined by

$f_1,f_2$1

with $f_1,f_2$2 and $f_1,f_2$3 tending to $f_1,f_2$4 through differentiability points. The stated properties are that the set-valued bracket is nonempty, compact, convex, and upper semicontinuous as a set-valued mapping, and that it reduces to the standard singleton bracket when the fields are smooth (Bardi et al., 2019).

Two elementary identities survive in set form: $f_1,f_2$5 and $f_1,f_2$6. What changes is that the output is no longer a single direction but a compact convex family of possible directions. This change is precisely what permits the bracket to remain meaningful under Rademacher-type almost-everywhere differentiability hypotheses (Motta et al., 2016).

2. Iterated brackets, asymptotics, and geometric meaning

The geometric role of Lie brackets persists in the set-valued setting: they encode directions generated by concatenations of flows that are not visible at first order. In the smooth case, a formal iterated Lie bracket $f_1,f_2$7 of length $f_1,f_2$8 enters asymptotic expansions of multi-flows through a term $f_1,f_2$9. In the nonsmooth setting, this expansion is replaced by a distance estimate: $[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$0 This estimate is the key mechanism by which set-valued iterated brackets retain operational content in control and boundary-regularity arguments (Bardi et al., 2019).

The corresponding nonsmooth Hörmander condition is formulated by selecting finitely many formal brackets $[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$1 and requiring that, at a point $[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$2, each underlying tuple of vector fields has the regularity needed for the bracket $[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$3, and that for arbitrary choices

$[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$4

one has

$[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$5

The “step” $[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$6 is the maximal bracket length among the $[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$7. Under symmetry of the control system, this condition yields small time local controllability and the estimate

$[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$8

near $[f_1,f_2]=Df_2\,f_1-Df_1\,f_2.$9 (Bardi et al., 2019).

This framework shows that set-valuedness does not eliminate bracket-generated geometry; it relocates it from a pointwise differential identity to an upper-semicontinuous family of limiting directions. A plausible implication is that the correct invariant in low-regularity settings is not a single commutator field but a compact convex bundle of bracket directions.

3. Degree-$$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$0 control Lyapunov functions

In nonlinear control, set-valued Lie brackets enter a generalized Lyapunov theory in which first-order dissipation is replaced by dissipation along iterated brackets. For a control system

$$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$1

the degree-$$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$2 Hamiltonian is defined by

$$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$3

where $$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$4 collects the formal iterated Lie brackets up to degree $$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$5, and for locally Lipschitz fields the values $$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$6 may be sets rather than singletons. A positive definite, proper, locally semiconcave function $$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$7 is a degree-$$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$8 control Lyapunov function if

$$[f_1,f_2]_{\rm set}(x):=\operatorname{co}\left\{\lim_{k\to\infty}[f_1,f_2](x_k):x_k\to x,\ x_k\in \mathrm{Diff}(f_1)\cap \mathrm{Diff}(f_2)\right\}.$$9

with $\mathrm{Diff}(f)$0 the set of limiting gradients (Motta et al., 2016).

The principal result is that the existence of a degree-$\mathrm{Diff}(f)$1 control Lyapunov function is sufficient for global asymptotic controllability to the target. This remains true when the iterated brackets appearing in the Hamiltonian are set-valued because the vector fields are only locally Lipschitz. The construction therefore enlarges the admissible descent directions beyond the original control vectors and allows Lyapunov inequalities to be enforced under weaker regularity assumptions than in the degree-1 theory (Motta et al., 2016).

Several examples in the cited work clarify why higher-degree and set-valued brackets are not merely technical embellishments. For the nonholonomic integrator

$\mathrm{Diff}(f)$2

the distance function fails to be a degree-1 control Lyapunov function, but it is a degree-2 control Lyapunov function because the bracket

$\mathrm{Diff}(f)$3

supplies the missing direction. A further example requires degree $\mathrm{Diff}(f)$4, showing that the filtration by bracket length is genuinely nontrivial. The nonsmooth examples are especially significant because classical brackets may be undefined at some points while the set-valued bracket remains available (Motta et al., 2016).

A common misconception is that set-valued brackets are relevant only when one gives up smoothness. The control-Lyapunov framework indicates a stronger point: even when smooth brackets exist, higher-degree bracket directions may be the decisive objects for asymptotic controllability.

4. Degenerate eikonal equations and viscosity regularity

Set-valued iterated Lie brackets also play a central role in the regularity theory of degenerate eikonal equations and associated minimum-time functions. The model equation discussed in the cited work is

$\mathrm{Diff}(f)$5

When the family $\mathrm{Diff}(f)$6 fails to span $\mathrm{Diff}(f)$7, coercivity is lost, and Lipschitz regularity need not hold. The set-valued bracket formalism provides sufficient conditions under which Hölder continuity can still be proved (Bardi et al., 2019).

The central boundary condition is transversality of a formal iterated bracket $\mathrm{Diff}(f)$8 of length $\mathrm{Diff}(f)$9 with the boundary normal $f$0: $f$1 Under suitable regularity assumptions on the target, this implies

$f$2

where $f$3 is the signed distance to the target. If every boundary point admits such a bracket of length at most $f$4, then the minimum-time function is locally $f$5-Hölder continuous. The same mechanism yields Hölder continuity of viscosity solutions of the Dirichlet problem (Bardi et al., 2019).

The work gives an explicit modulus of continuity: $f$6 This formula links the regularity of the solution to the regularity of the running cost and boundary data together with the bracket step. It also clarifies that the bracket length controlling transversality is the quantity that sets the Hölder exponent.

The examples emphasize the geometric content of the theory. For the Heisenberg system, a step-$f$7 condition yields local $f$8-Hölder continuity. For Grushin-type fields, a step-$f$9 condition yields local $\operatorname{co}$0-Hölder continuity. The same source also proves that, at least for smooth vector fields and target, the sufficient conditions are essentially necessary in the second-order case: local $\operatorname{co}$1-Hölder regularity at a boundary point is equivalent to the presence of either a bracket of length $\operatorname{co}$2 pointing inward or an appropriate second-order tangential condition (Bardi et al., 2019).

5. Noncommutative geometry and projector-dependent brackets

A distinct form of set-valued Lie bracket appears in noncommutative geometry. There the issue is not lack of differentiability but the absence of a canonical antisymmetrization for compositions of noncommutative vector fields. In this setting, vector fields are defined as

$\operatorname{co}$3

where $\operatorname{co}$4 is a unital associative algebra equipped with a differential calculus. Symbols are defined by

$\operatorname{co}$5

and a substantial part of the theory is devoted to representability of differential operators by jet modules and to the symbol calculus needed to control compositions (Flood et al., 2023).

The bracket construction proceeds by identifying an antisymmetrized submodule $\operatorname{co}$6, obtained as the annihilator of the quantum symmetric forms. Under the condition

$\operatorname{co}$7

Theorem 8.7 states that for any

$\operatorname{co}$8

the composition

$\operatorname{co}$9

is again a vector field. Given a projector

$k$0

the corresponding quantum Lie bracket is defined by

$k$1

The construction ensures closure, but the bracket depends on the choice of $k$2; in that sense it is set-valued unless additional structure supplies a canonical antisymmetrizer (Flood et al., 2023).

This version of set-valuedness differs conceptually from the nonsmooth-control version. In the latter, multivaluedness comes from taking limit points at nearby differentiability points; in the former, it comes from nonuniqueness of antisymmetrization in a noncommutative tensor calculus. The comparison suggests that “set-valued Lie bracket” is not a single universal construction, but a family of remedies for the failure of classical uniqueness under different structural obstructions.

6. Deformations, compatible brackets, and related generalizations

In deformation theory for Lie algebroids, set-valued brackets appear in a further generalized sense. A first-order deformation of a bracket is written

$k$3

where $k$4 is a $k$5-cocycle and may be viewed as “up to homotopy.” The cited work frames such deformations through representations up to homotopy, proves a functorial differentiation procedure from Lie groupoids to Lie algebroids, and establishes a van Est type isomorphism in that setting. Under the hypothesis that a Lie algebroid $k$6 admits a proper integrating Lie groupoid with $k$7-connected source fibers, the deformation cohomology satisfies

$k$8

so infinitesimal deformations of the bracket are trivial under those geometric conditions (Abad et al., 2010).

A broader adjacent literature studies structures with multiple, higher, or set-based brackets rather than set-valuedness in the nonsmooth or noncommutative sense. In interchange algebras with associative operations $k$9 and $[ [f_1,f_2],f_3]$0, the polarized brackets

$[ [f_1,f_2],f_3]$1

are both Lie brackets, and their interaction yields new polynomial identities in degrees $[ [f_1,f_2],f_3]$2 and $[ [f_1,f_2],f_3]$3 beyond anticommutativity and Jacobi (Bremner et al., 2014). For a pair of compatible Lie brackets $[ [f_1,f_2],f_3]$4 and $[ [f_1,f_2],f_3]$5, every linear combination

$[ [f_1,f_2],f_3]$6

is again a Lie bracket, and the universal enveloping algebra of an $[ [f_1,f_2],f_3]$7-dimensional compatible Lie algebra has growth rate $[ [f_1,f_2],f_3]$8 (Gubarev, 2021). Higher-arity analogues also occur: complete generalized Wronskians define $[ [f_1,f_2],f_3]$9-ary strong homotopy Lie brackets with

$f_1,f_2\in C^{1,1}$0

producing explicit finite-dimensional polynomial SH-Lie algebras in special cases (Ķēniņš et al., 26 May 2026). A different combinatorial direction defines a Lie bracket on partitions of finite sets by antisymmetrizing insertion-based composition, with antisymmetry and Jacobi identity established at the level of partitions themselves (Zhou, 2021).

Setting	Source of set-valuedness or multiplicity	Role
Locally Lipschitz vector fields	Convex hull of nearby bracket limits	Controllability and Hölder regularity
Noncommutative geometry	Dependence on projector $f_1,f_2\in C^{1,1}$1	Closure of a bracket on vector fields
Deformation theory	Brackets controlled up to homotopy	Cohomology and rigidity
Compatible or multiple brackets	More than one bracket operation	Operadic and polynomial identities
$f_1,f_2\in C^{1,1}$2-ary or set-based brackets	Higher arity or combinatorial insertion	SH-Lie and graph/partition structures

Taken together, these developments show that set-valued Lie brackets belong to a wider reconfiguration of Lie theory in which uniqueness of the binary commutator is no longer assumed a priori. In one direction, the bracket becomes a compact convex family of limiting commutators; in another, it becomes projector-dependent; in another, it is replaced by compatible pairs or $f_1,f_2\in C^{1,1}$3-ary operations. The common thread is that the classical bracket is being reformulated so that its geometric or algebraic consequences survive in settings where smooth single-valued commutators are too rigid to exist.