Measure-theoretic Lie brackets for nonsmooth vector fields

In this paper we prove a generalization of the classical notion of commutators of vector fields in the framework of measure theory, providing an extension of the set-valued Lie bracket introduced by Rampazzo-Sussmann for Lipschitz continuous vector fields. The study is motivated by some applications to control problems in the space of probability measures, modeling situations where the knowledge of the state is probabilistic, or in the framework of multi-agent systems, for which only a statistical description is available. Tools of optimal transportation theory are used.

1. Introduction. In [17] the authors give a generalization of the classical notion of Lie bracket (or commutator ) of two smooth vector fields X, Y , in order to study the commutativity of the flows of two vector fields basically just assuming that the flows are well-defined (e.g., the two vector fields are locally Lipschitz continuous). In this framework, the classical Lie bracket [X, Y ](·) appears to be defined only a.e. w.r.t. Lebesgue measure, moreover, as showed with many examples in [17], even at the points where it can be defined, it does not catch all the local features of the two flows.
By means of a suitable construction, in [17] the authors define an object, called set-valued Lie bracket, which associates to every point of the space a suitable set [X, Y ] set (·), which in the classical smooth case is reduced to the usual Lie bracket, and turns out to be the convex hull of the upper Kuratowski limit of the classical Lie brackets (which are defined in a Lebesgue full measure subset, in particular in a dense subset).
They also prove that the basic properties enjoyed by the classical Lie bracket, have their natural counterparts. More precisely, if X, Y are locally Lipschitz vector fields on a manifold M of class C 2 , denoted by φ X t and φ Y t their flows at time t, it is proved that 1. Asymptotic formula: for all q ∈ M we have , [X, Y ] set (q) = 0.
2. Commutativity of the flows: 3. Simultaneous flow-box theorem: if X i , i = 1, . . . , d are locally Lipschitz vector fields on a d-dimensional manifold M of class C 2 satisfying [X i , X j ] set (q) = {0} for all q ∈ M i, j = 1, . . . , d then, around every point q ∈ M where the vector fields are independent, there exists a Lipschitz change of coordinates with Lipschitz inverse, sending X i to the i-th element of the canonical basis e i of T q M . The main ingredient to prove the results of [17] is an exact integral formula Lemma 4.5 of [17]). In this context, the term exact is used in opposition to asymptotic. This integral formula turns out very useful to be handled, and, together with a regularization argument, yields all the main results of the paper.
In [16], these results are applied to give a nonsmooth version of the Frobenius theorem for Lipschitz distributions of vector fields on a manifold. The generalization of the construction of [17] to higher order Lie bracket is not straigthforward, as pointed out in Section 7 of [17], and has been recently proved in the two papers [10], which generalized the exact formula for the single Lie bracket to general nested brackets, and the recent [11].
To make the computations, the authors in [17] make extensively use of the Agrachev -Gamkrelidze formalism (AGF), introduced by Agrachev and Gamkrelidze in the '70s. The main idea of this formalism is to embed all the main objects of the flow analysis in a convenient subspace of the space of distributions, using the linear structure of the latter to perform all the computations. In this setting, each point q is represented by a Dirac delta δ q , a vector field is seen as a differential operator on C ∞ c functions, and the flow of vector fields is seen as the push forward operator. With the AGF formalism, all the computations turn out to be shortned and simplified (see e.g. Section 2 of [17] for an outline of the AGF formalism, containing also a rigorous justification and some examples).
It is well known that, in the classical framework, the vector space Lie(F ) generated by all the vector fields built from a given set F of vector fields by means of possibly nested Lie bracket, is deeply related to controllability properties of the finite-dimensional driftless control-affine systems where the controlled vector fields are the element of F . Roughly speaking, Lie bracket operations enlarge the set of admissible displacements that a particle can reach in a given amount of time by following the admissible trajectories of the system, even if, in general, a Lie bracket does not give an admissible direction for the system.
Hence, the study of higher order conditions for attainability plays an important role. In the classical finite-dimensional setting, Petrov's condition represents a first order requirement on the trajectory and can be interpreted as the request that for each point sufficiently near to the target there exists an admissible trajectory which points sufficiently towards the target at the first order, indeed it involves the first order term of at least one admissible trajectory, i.e. an admissible velocity. Since it is a strong condition to be satisfied, it is natural to look for higher order conditions when the first one does not hold, by involving higher order terms of the expansion of the trajectory. It has been studied (see [12]) that these conditions involve Lie bracket of admissible vector fields and they can be viewed as Petrov's conditions of higher order.
In recent papers [4], [5], [6], [7], [8], [9], some control problems in the space of probability measures on R d are studied, as a natural generalization of control problems in finite-dimensional space when the initial state is known only up to some uncertainty, or to model situations where the number of agents is so huge to make viable only a statistical (macroscopic) description of the system. In the first case, the time-evolving measure represents our probabilistic knowledge about the state of the particle, while, in the second case, it represents the statistical distribution of the agents.
In all these problems, the dynamics is given by a controlled continuity equation, to be satisfied in the sense of distributions, where the current density (which is the control in the problem) is chosen among the Borel selections of a given set-valued map, which can be seen as the underlying microscopic classical dynamics, i.e., the dynamics followed by each agent. In order to study controllability problems in this framework, it turns out to be a natural problem to define some correspondent quantity for the Lie bracket in a measure-theoretic setting by using tools of transport theory. The study of controllability conditions involving measure-theoretic Lie bracket is still an open problem in this setting. We refer the reader to [13,14] for the study of sufficient conditions granting small time-local attainability in finitedimension.
Our strategy can be summarized as follows: by exploiting the main idea of the AGF formalism, instead of considering Dirac deltas, we consider probability measures on R d , and define our object as limit (in a suitable topology) of an asymptotic formula like the one considered by Rampazzo-Sussmann, but instead of the evaluation at the point q, corresponding to the choice of δ q , we consider the push forward of a probability measure µ along the flow. Under suitable assumptions, we are able to consider the convexified Kuratowski upper limit of this construction as in [17], thus defining a set-valued measure theoretic Lie bracket, which -by constructionsatisfies the asymptotic formula and the commutativity property. We notice that this object, being a set of vector-valued measures absolutely continuous w.r.t. µ, has no longer a pointwise meaning, unless the starting measure is purely atomic.
We give also some representation formula, which allows to compare our results with the results of [17], showing that in the case of Dirac deltas, the two constructions agree and, slightly more generally, under the Lipschitz assumptions of [17], the density of each element w.r.t. a general probability measure µ is an L p µ -selection of the set-valued Lie bracket defined in [17].
The paper is structured as follows: in Section 2 we review some preliminaries of measure theory and of differential geometry, in Section 3 we introduce the main objects of our study and formulate the main results, in Section 4 we compare our result with the construction of Rampazzo-Sussmann. We conclude providing an example illustrating our construction in Section 5.

Preliminaries and notation.
2.1. Measure theory. Our main reference for this part is [2].
Given T > 0, we set Γ T = C 0 ([0, T ]; R d ), which is a separable Banach space when endowed with the usual sup norm. We denote by e t : R d × Γ T → R d the evaluation operator, defined as e t (x, γ) = γ(t). Notice that e t (x, γ) does not depend on x.

GIULIA CAVAGNARI AND ANTONIO MARIGONDA
Given a family of Banach spaces {X i } i∈I , we define the Borel maps r i : We denote with Id R d the identity map on R d . Given a complete and separable metric space X, let us denote by P(X) the space of Radon probability measures on X. We have that P(X) = (C 0 b (X)) and that the w * -topology on P(X) induced by this duality is metrizable (for instance by the Prokhorov metric). We will denote by d P any metric on P(X) inducing the w *topology on P(X). We will denote by M (R d ; R k ) the set of vector-valued Radon If X is a separable metric space, we will denote with Bor(X) the set of Borel maps from X to R and with Bor b (X) the set of bounded Borel maps from X to R.
For the following, let X be a separable Banach space.
Definition 2.1 (Wasserstein distance). Given µ 1 , µ 2 ∈ P(X), p ≥ 1, we define the p-Wasserstein distance between µ 1 and µ 2 by setting where the set of admissible transport plans Π(µ 1 , µ 2 ) is defined by Definition 2.2 (p-moment). Let µ ∈ P(X), p ≥ 1. We say that µ has finite p-moment if Equivalently, we have that µ has p-moment finite if and only if for every x 0 ∈ X we have We denote by P p (X) the subset of P(X) consisting of probability measures with finite p-moment.
2. the set K has uniformly integrable p-moments, p ≥ 1, if |x| p is uniformly integrable with respect to K . 3. if K = {µ n } n∈N ⊆ P(X), p ≥ 1, µ n * µ ∈ P(X), the set K has uniformly integrable p-moments if and only if for every continuous function f : R d → R such that there exist a, b ≥ 0 and Proposition 1. P p (X) endowed with the p-Wasserstein metric W p (·, ·) is a complete separable metric space. Moreover, given a sequence {µ n } n∈N ⊆ P p (X) and µ ∈ P p (X), we have that the following are equivalent 2. µ n * µ and {µ n } n∈N has uniformly integrable p-moments.
Definition 2.4 (Formal bracket). We denote by Diffeo(R d ) the set of all diffeomorphisms of R d . Let ψ, ϕ ∈ Diffeo(R d ) be two diffeomorphisms. We define their formal bracket by setting: Since for every ψ, ϕ ∈ Diffeo(R d ) we have that [ψ, ϕ] ∈ Diffeo(R d ), by iterating the procedure we can construct formal bracket expressions by nesting formal brackets of diffeomorphisms. Given a subset S ⊆ Diffeo(R d ), we define the length (also order or depth) of nested formal brackets of elements of S by induction. If ϕ ∈ S is a single diffeomorphism, then ord (ϕ) = 1. Otherwise, if A and B are formal bracket expressions of elements of S , we set ord [A, B] = ord A + ord B.
For t sufficiently small, it is well known that φ X t (·) is a diffeomorphism. Given two C 1 -smooth vector fields X, Y , we have that where on the right hand side we have the usual Lie bracket of vector fields defined in local coordinates by: The correspondence between the first nonvanishing derivative at 0 of flows generating the bracket and the order of the Lie bracket is explained in the following classical result (see e.g., Theorem 1 in [15]). Theorem 2.6. Let k ∈ N\{0, 1}, M be a manifold of class C k , and for i = 1, . . . , k where Vec k (M ) is the set of vector fields on M of class C k . Then for each formal bracket expression B of order k (w. where the last expression is computed substituting each φ i t with X i in B(φ 1 t , . . . , φ k t ), and then computing the nested Lie brackets of vector fields.
3. Measure-theoretic Lie bracket. In this section we introduce the basic objects of our analysis, proving also the main results of the paper.
where cl d P denotes the closure in the w * -topology. If K = P(R d ) we will omit the subscript K.
Define the measures Remark 1. The main motivation for considering a general subset K of P(R d ) comes from applications, where for example we are able to measure only averaged quantities w.r.t. Lebesgue's measure.
We will now provide some estimates on the p-moments of the measures η Ψ K µ and π Ψ K ,m µ,t associated to Ψ K .

If
Proof.

If there exists a Borel map
We define now a measure-theoretic object related to the limit of where cl Wp denotes the closure in the W p -topology, and π Ψ K ,m µ ,t is defined as in Definition 3.1.
The set of vector-valued measures {V µ : V ∈ V p m (µ, Ψ K )} will be the object generalizing the asymptotic behaviour of the vector-valued measure Ψ t − Id R d t m µ , in the sense precised below.
in the sense of distributions.
Hence we have Taking the limit for i → +∞, and recalling that m p (π Ψ K ,m ti,µ (i) ) is uniformly bounded which concludes the proof by the arbitrariness of ϕ ∈ C ∞ c (R d ).
Corollary 1. In the same assumptions of Lemma 3.4, assume that Proof. The result comes immediately, since we have and in both cases the right hand side tends to 0 by assumption.
We are going to provide now a sufficent condition ensuring that the above defined sets are nonempty.  More precisely, if π = µ ⊗ σ π x ∈ P p m (µ, Ψ K ) then the map defined as 1. Given π ∈ P p m (µ, Ψ K ) as in the statement, we estimate the L p µ -norm of V (·) by applying Jensen's inequality Then we have that V ∈ V p m (µ, Ψ K ), which turns out to be nonempty. The converse is trivial. 2. Let {µ (i) } i∈N be a sequence in K, {t i } i∈N ⊆]0, T ] be such that Since W p (µ (i) , µ) → 0, we have that there exists C > 0 such that m 1/p p (µ (i) ) ≤ C for all i ∈ N. Define π Ψ K ,m µ (i) ,ti as in Definition 3.1, and notice that, by assumption, for i sufficiently large we have ≤ C + 1. Thus, according to Lemma 3.2 item (1), In particular, according to Remark 5.1.5 in [2], up to passing to a subsequence, we can assume that there exists π ∞ ∈ P p (R d × R d ) such that W p (π Ψ K ,m µ (i) ,ti , π ∞ ) → 0, yielding π ∞ ∈ P p m (µ, Ψ K ) and m p (π ∞ ) ≤ (C + C + 1) p . To conclude, it is enough to apply the previous item.

Then we have
Proof. The inclusion ⊇ holds trivially true. We prove the converse inclusion. Let π ∈ P p m (µ, Ψ K ), in particular there exists since ϕ W ≡ 1 on supp µ. Thus we have µ (i) W * µ for all 0 < δ < T . For any 0 < δ < T we have This implies m p (π Ψ K ,m µ (i) W ,ti ) ≤ 2m p (π Ψ K ,m µ (i) ,ti ), for all i ≥ i δ , by Monotone Convergence Theorem. Since by assumption W p (π, π Ψ K ,m µ (i) ,ti ) → 0, we have that m p (π Ψ K ,m µ (i) ,ti ) is uniformly bounded, and so, up to passing to a non relabeled subsequence, we have that there exists π ∈ P p (R d × R d ) such that W p (π , π Ψ K ,m µ (i) W ,ti ) → 0 as i → +∞. To prove that π = π , which will conclude the proof by the arbitrariness of W and δ, it is enough to show that d P (π, π Ψ K ,m and so W p (π, π Ψ K ,m ti,µ We will now provide some representation formulas for the function on V p m (µ, Ψ K ), proving also some refinement under additional assumptions. These will be used to establish a comparison with the set-valued Lie brackets defined by Rampazzo-Sussmann in [17].
Proof. Let V ∈ V p m (µ, Ψ K ). There exist sequences {t i } i∈N ⊆]0, T ], t i → 0 + and {µ (i) } i∈N ⊆ K, µ (i) * µ, and a family of probability measures {ξ x } x∈R d uniquely defined for µ-a.e. x ∈ R d such that denoted by π := µ⊗ξ x , we have W p (π Ψ K ,m µ (i) ,ti , π) → 0 and For any σ ∈]0, T ] we define a set-valued map G σ : R d ⇒ R d by taking Notice that dom G σ ⊇ D. This set-valued map has closed graph, indeed, let {x n } n∈N , {y n } n∈N ⊆ R d , x, y ∈ R d be such that x n → x, y n → y, y n ∈ G σ (x n ) for all n ∈ N. Fix δ > 0 and let n δ > 0 be such that |x n − x| < δ for all n ≥ n δ . For every δ > 0 and n ≥ n δ we have that y n ∈ co S σ,δ m,D (x n ) ⊆ co S σ,δ +|xn−x| m,D (x) ⊆ co S σ,δ +δ m,D (x).
By passing to the limit as n → +∞ we have y ∈ co S σ,δ +δ m,D (x) for all δ , δ > 0, and then by taking the intersection on δ, δ > 0 we have y ∈ G σ (x).
By Jensen's inequality we have Recalling Lemma 5.1.7 in [2], by l.s.c. of g σ (·, ·) we have 4. Application to the composition of flows of vector fields. As seen in the Introduction, in [17] the authors extended the definition of a Lie bracket of two C 1 vector fields to the case of two Lipschitz continuous vector fields X, Y , that is an assumption implying continuity of Ψ t (·) := [φ X t , φ Y t ](·). In this case, the Lie bracket of the vector fields at every point turns out to be a set. Moreover, they provided in this framework an asymptotic formula for the flows and the generalization of other classical results holding for the Lie bracket of vector fields.
A natural question is to compare our construction with the one in [17] when the starting measure is reduced to a Dirac delta, in the spirit of the AGF formalism. The aim of this section is to perform such a comparison, showing that -roughly speaking -the density V of the measure theoretic bracket V µ is a L p µ -selection of the Rampazzo-Sussmann set-valued Lie bracket. In particular, when µ = δ q , the two constructions are reduced to the same object.
We will take K = P(R d ) throughout the section, hence we will omit the condition (D 4 ) in Definition 3.1 since it follows from (D 1 ).
We recall the following definition from [17]. where dom(Df ) and dom(Dg) denotes the set of differentiability points of f and g, respectively. Recalling Rademacher's Theorem, when f is Lipschitz continuous it is differentiable at a.e. x ∈ R d , thus dom(Df ) ∩ dom(Dg) has full measure in R d .
According to Remark 3.6 in [17], the following equivalent definition can be given Recall that in general ∂(f × g)(x) ⊆ ∂f (x) × ∂g(x), and the inclusion may be strict. We can recast the above definition by  According to the representation formula, we have that if V p 2 (µ, Ψ) = ∅, we must have V (x, y) = x − y x 2/3 y 2/3 (1, 1), for µ-a.e. (x, y) ∈ D and all V ∈ V p 2 (µ, Ψ).