Complexity of control-affine motion planning
aa r X i v : . [ m a t h . O C ] D ec COMPLEXITY OF CONTROL-AFFINE MOTION PLANNING
F. JEAN ∗ AND D. PRANDI †‡ Abstract
In this paper we study the complexity of the motion planning problem for control-affinesystems. Such complexities are already defined and rather well-understood in the particular caseof nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and resultsto systems with a drift. Accordingly, we present various definitions of complexity, as functions ofthe curve that is approximated, and of the precision of the approximation. Due to the lack of time-rescaling invariance of these systems, we consider geometric and parametrized curves separately.Then, we give some asymptotic estimates for these quantities.
Key words: control-affine systems, sub-Riemannian geometry, motion planning, complexity.
Introduction
The concept of complexity was first developed for the non-holonomic motion planning problemin robotics. Given a control system on a manifold M , the motion planning problem consists infinding an admissible trajectory connecting two points, usually under further requirements, such asobstacle avoidance. If a cost function is given, it makes sense to try to find the trajectory costingthe least.Different approaches are possible to solve this problem (see [24]). Here we focus on those basedon the following scheme:(1) find any (usually non-admissible) curve or path solving the problem,(2) approximate it with admissible trajectories.The first step is independent of the control system, since it depends only on the topology of themanifold and of the obstacles, and it is already well understood (see [28]). Here, we are interestedin the second step, which depends only on the local nature of the control system near the path.The goal of the paper is to understand how to measure the complexity of the approximation task.By complexity we mean a function of the non-admissible curve Γ ⊂ M (or path γ : [0 , T ] → M ),and of the precision of the approximation, quantifying the difficulty of the latter by means of thecost function. ∗ ENSTA ParisTech, 828 Boulevard des Mar´echaux 91762 Palaiseau, France, and Team GECO,INRIA Saclay – ˆIle-de-France, † Laboratoire LSIS, Universit´e de Toulon, France, and Team GECO, INRIA-Centre de RechercheSaclay, ‡ SISSA, Trieste, Italy
E-mail addresses : [email protected], [email protected] .This work was supported by the European Research Council, ERC StG 2009 “GeCoMethods”, contract number239748; by the ANR project GCM , program “Blanche”, project number NT09 504490; and by the Commission of theEuropean Communities under the 7th Framework Programme Marie Curie Initial Training Network (FP7-PEOPLE-2010-ITN), project SADCO, contract number 264735. .1. Control theoretical setting.
In this paper, we consider a control-affine system on a smoothmanifold M is a control system in the form(1) ˙ q ( t ) = f ( q ( t )) + m X i =1 u i ( t ) f i ( q ( t )) , a.e. t ∈ [0 , T ] , where u : [0 , T ] → R m is an integrable control function and f , f , . . . , f m are (not necessarilylinearly independent) smooth vector fields. The uncontrolled vector field f is called the drift .These kind of systems appear in plenty of applications. As examples we cite: mechanical systemswith controls on the acceleration (see e.g., [9, 5]), where the drift is the velocity, quantum control (seee.g., [10, 8]), where the drift is the free Hamiltonian, or the swimming of microscopic organisms(see, e.g., [27]). We always assume the strong H¨ormander condition, i.e., that the iterated Liebrackets of the controlled vector fields f , . . . , f m generate the whole tangent space at any point.This guarantees the small time local controllability of system (1) and allows us to associate to(1) a sub-Riemannian control system. Such assumption is generically satisfied, e.g., by finite-dimensional quantum control systems with two controls, as the ones studied in [6, 7, 11]. Exceptwhen explicitly stated, we do not make any assumption on the dimension of span { f ( q ) , . . . , f m ( q ) } which, in particular, can depend on the point q ∈ M .When posing f = 0 in (1) we obtain the (small) sub-Riemannian control system associated with(1), i.e., the driftless control system in the form(2) ˙ q ( t ) = m X i =1 u i ( t ) f i ( q ( t )) , a.e. t ∈ [0 , T ] , This system satisfies the H¨ormander condition is satisfied, i.e., the iterated Lie brackets of thevector fields f , . . . , f m generate the whole tangent space at any point. Moreover, we will alwaysassume the sub-Riemannian structure to be equiregular (see Section 2.1). Given a sub-Riemanniancontrol system, a natural choice for the cost is the L -norm of the controls. Due to the linearityand the reversibility in time of such a system, the associated value function is in fact a distance,called Carnot-Carath´eodory distance, that endows M with a metric space structure.Our work will focus on the following cost functions,(3) J ( u, T ) = Z T vuut m X j =1 u j ( t ) dt and I ( u, T ) = Z T vuut m X j =1 u j ( t ) dt. Namely, J ( u, T ) = k u k L ([0 ,T ] , R m ) and I ( u, T ) = k (1 , u ) k L ([0 ,T ] , R m +1 ) . Let q ∈ M and define q u : [0 , T ] → M as the trajectory associated with a control u ∈ L ([0 , T ] , R m ) such that q u (0) = q .The cost J , measuring the L -norm of the control, quantifies the cost spent by the controller tosteer the system (1) along q u . On the other hand, I measures the Riemannian length of q u withrespect to a Riemannian metric such that f , f , . . . , f m are orthonormal.Fix a time T > J ( q, q ′ ) and V I ( q, q ′ ) as the infimaof the costs J and I , respectively, over all controls u ∈ U T = S
0. As abyproduct of this choice, by taking T sufficiently small, it is then possible to prevent any exploitationof the geometry of the orbits of the drift (that could be, for example, closed). Let us also remarkthat, since the controls can be defined on arbitrarily small times, it is possible to approximateadmissible trajectories for system (1) via trajectories for the sub-Riemannian associated system(i.e., the one obtained by posing f ≡
0) rescaled on small intervals.1.2.
Complexities.
Heuristically, the complexity of a curve Γ (or path γ : [0 , T ] → M ) at precision ε is defined as the ratio(4) “cost” to track Γ at precision ε “cost” of an elementary ε -piece . In order to obtain a precise definition of complexity, we need to give a meaning to the notionsappearing above. Namely, we have to specify what do we mean by “cost” , tracking at precision ε ,and elementary ε -piece. Indeed, these choices will depend on the type of motion planning problemat hand.First of all, we classify motion planning problems as time-critical or static , depending on wetherthe constraints depend on time or not. The typical example of static motion planning problemis the obstacle avoidance problem with fixed obstacles. On the other hand, the same problemwhere the position of the obstacles depends on time, or the rendez-vous problems, are examples oftime-critical motion planning problems.For static motion planning problems, the solution of the first step of the motion planning scheme(introduced at the beginning of the paper) is usually given as a curve, i.e., a dimension 1 connectedsubmanifolds of Γ ⊂ M diffeomorphic to a closed interval. On the other hand, in time-criticalproblems we have to keep track of the time. Thus, for this type of problems, the solution of thefirst step is a path, i.e., a smooth injective function γ : [0 , T ] → M . As a consequence, whencomputing the complexity of paths we will require the approximating trajectories to respect alsothe parametrization, and not only the geometry, of the path. While in the sub-Riemannian case,due to the time rescaling properties of the control system, these concepts coincide, this is not thecase for control-affine systems.In this work, we consider four distinct notions of complexity, two for curves (static problems)and two for paths (time-critical problems). In both cases, one of the two will be based on theinterpolation of the given curve or path, while the other will consider trajectories that stays nearthe curve or path. Thus, for this complexity, we will need to fix a metric. In this work we willconsider only the sub-Riemannian metric of the associated sub-Riemannian control system (2)).We remark that the two complexity for curves are the same as the sub-Riemannian ones alreadyintroduced in [20, 21]. This is true also for what we call the neighboring approximation complexity ofa path, since in the sub-Riemannian case it coincides with the tubular approximation complexity.On the other hand, what we call the interpolation by time complexity never appeared in theliterature, to our knowledge. Here we give the definitions for a generic cost J : U T → [0 , + ∞ ).Fix a curve Γ and, for any ε >
0, define the following complexities for Γ. • Interpolation by cost complexity: (see Figure 1) For ε >
0, let an ε -cost interpolation ofΓ to be any control u ∈ U T such that there exist 0 = t < t < . . . < t N = T ≤ T forwhich the trajectory q u with initial condition q u (0) = x satisfies q u ( T ) = y , q u ( t i ) ∈ Γ and J ( u | [ t i − ,t i ) , t i − t i − ) ≤ ε , for any i = 1 , . . . , N . Then, letΣ int (Γ , ε ) = 1 ε inf (cid:8) J ( u, T ) | u is an ε -cost interpolation of Γ (cid:9) . The cost appearing in (4) is not necessarily related with the cost function ( J or I ) taken into account. This isthe reason for the quotation marks. Γ yq u ( t i − ) q u ( t i ) ≤ ε Figure 1.
Interpolation bycost complexity x Γ y Tube(Γ , ε ) q u ( · ) Figure 2.
Tubular approxi-mation complexity x yq u ( t i − ) = γ ( t i − ) q u ( t i ) = γ ( t i )on avg. ≤ ε Figure 3.
Time interpola-tion complexity x yq u ( t ) γ ( t )B SR ( γ ( t ) , ε ) Figure 4.
Neighboring ap-proximation complexityThis function measures the number of pieces of cost ε necessary to interpolate Γ. Namely,following a trajectory given by a control admissible for Σ int (Γ , ε ), at any given moment itis possible to go back to Γ with a cost less than ε . • Tubular approximation complexity: (see Figure 2) Let Tube(Γ , ε ) to be the tubular neigh-borhood of radius ε around the curve Γ w.r.t. the small sub-Riemannian system associatedwith (1) (obtained by putting f ≡
0, see Section 2.2), and defineΣ app (Γ , ε ) = 1 ε inf J ( u, T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < T ≤ T ,q u (0) = x, q u ( T ) = y,q u (cid:0) [0 , T ] (cid:1) ⊂ Tube(Γ , ε ) This complexity measures the number of pieces of cost ε necessary to go from x to y staying inside the sub-Riemannian tube Tube(Γ , ε ). Such property is especially useful formotion planning with obstacle avoidance. In fact, if the sub-Riemannian distance of Γ fromthe obstacles is at least ε >
0, then trajectories obtained from controls admissible forΣ app (Γ , ε ), ε < ε , will avoid such obstacles.We then define the following complexities for a path γ : [0 , T ] → M at precision ε > • Interpolation by time complexity: (see Figure 3) Let a δ -time interpolation of γ to be anycontrol u ∈ L ([0 , T ] , R m ) such that its trajectory q u : [0 , T ] → M in (1) with q u (0) = γ (0)is such that q u ( T ) = γ ( T ) and that, for any interval [ t , t ] ⊂ [0 , T ] of length t − t ≤ δ ,there exists t ∈ [ t , t ] with q u ( t ) = γ ( t ). Then, fix a δ > σ int ( γ, ε ) = inf (cid:26) Tδ (cid:12)(cid:12)(cid:12)(cid:12) δ ∈ (0 , δ ) and exists u ∈ L ([0 , T ] , R m ), δ -time interpolation of γ , s.t. δ J ( u, T ) ≤ ε (cid:27) . Controls admissible for this complexity will define trajectories such that the minimal averagecost between any two consecutive times such that γ ( t ) = q u ( t ) is less than ε . It is thus well uited for time-critical applications where one is interested in minimizing the time betweenthe interpolation points - e.g. motion planning in rendez-vous problem. • Neighboring approximation complexity: (see Figure 4) Let B SR ( p, ε ) denote the ball of radius ε centered at p ∈ M w.r.t. the small sub-Riemannian system associated with (1) (obtainedby putting f ≡
0, see Section 2.2), and define σ app ( γ, ε ) = 1 ε inf (cid:26) J ( u, T ) (cid:12)(cid:12)(cid:12)(cid:12) q u (0) = x, q u ( T ) = y,q u ( t ) ∈ B SR ( γ ( t ) , ε ) , ∀ t ∈ [0 , T ] (cid:27) . This complexity measures the number of pieces of cost ε necessary to go from x to y following a trajectory that at each instant t ∈ [0 , T ] remains inside the sub-Riemannian ballB SR ( γ ( t ) , ε ). Such complexity can be applied to motion planning in rendez-vous problemswhere it is sufficient to attain the rendez-vous only approximately, or to motion planningwith obstable avoidance where the obstacles are moving.Whenever we will need to specify with respect to which cost a complexity is measured, we willwrite the cost function as apex – e.g., to we will denote the interpolation by cost complexity w.r.t. J as Σ int J .We remark that for the interpolation by time complexity the “cost” in (4) is the time, while forall the other complexities it is the cost function associated with the system. For the motivation ofthe bound on δ in the definition of the interpolation by time complexity, see Remark 5.3. Finally,whenever a metric is required, we use the sub-Riemannian one. Although such metric is natu-ral for control-affine systems satisfying the H¨ormander condition, nothing prevents from definingcomplexities based on different metrics.Two functions f ( ε ) and g ( ε ), tending to ∞ when ε ↓ weakly equivalent (denoted by f ( ε ) ≍ g ( ε )) if both f ( ε ) /g ( ε ) and g ( ε ) /f ( ε ), are bounded when ε ↓
0. When f ( ε ) /g ( ε ) (resp. g ( ε ) /f ( ε ))is bounded, we will write f ( ε ) g ( ε ) (resp. f ( ε ) < g ( ε )). In the sub-Riemannian context, thecomplexities are always measured with respect to the L cost of the control, J . Then, for any curveΓ ⊂ M and path γ : [0 , T ] → M such that γ ([0 , T ]) = Γ it holds Σ J int (Γ , ε ) ≍ Σ J app (Γ , ε ) ≍ σ J app ( γ, ε ).A complete characterization of weak asymptotic equivalence of sub-Riemannian complexities isobtained in [23]. We state here such characterization in the special case where { f , . . . , f m } definesan equiregular structure. Theorem 1.1.
Assume that { f , . . . , f m } defines an equiregular sub-Riemannian structure. Let Γ ⊂ M be a curve and γ : [0 , T ] → M be a path such that γ ([0 , T ]) = Γ . Then, if there exists k ∈ N such that T q Γ ⊂ ∆ k ( q ) \ ∆ k − ( q ) for any q ∈ Γ , it holds Σ int (Γ , ε ) ≍ Σ app (Γ , ε ) ≍ σ app ( γ, ε ) ≍ ε k . Here the complexities are measured w.r.t. the cost J ( u, T ) = k u k L ([0 ,T ] , R m ) . We mention also that for a restricted set of sub-Riemannian systems, i.e., one-step bracketgenerating or with two controls and dimension not larger than 6, strong asymptotic estimates andexplicit asymptotic optimal syntheses are obtained in the series of papers [26, 15, 16, 17, 18, 19, 14](see [13] for a review).1.3.
Main results.
Our first result is the following. It completes the description of the sub-Riemannian weak asymptotic estimates started in Theorem 1.1, describing the case of the interpo-lation by time complexity. It is proved in Section 5.
Theorem 1.2.
Assume that { f , . . . , f m } defines an equiregular sub-Riemannian structure and let γ : [0 , T ] → M be a path. Then, if there exists k ∈ N such that ˙ γ ( t ) ∈ ∆ k ( γ ( t )) \ ∆ k − ( γ ( t )) for ny t ∈ [0 , T ] , it holds σ int ( γ, ε ) ≍ ε k . Here the complexity is measured w.r.t. the cost J ( u, T ) = k u k L ([0 ,T ] , R m ) . Since in the sub-Riemannian context one is only interested in the cost J , Theorems 1.1 and1.2 completely characterize the weak asymptotic equivalences of complexities of equiregular sub-Riemannian manifolds.The main result of the paper is then a weak asymptotic equivalence of the above defined com-plexities in control-affine systems, generalizing Theorems 1.1 and 1.2. Theorem 1.3.
Assume that { f , . . . , f m } defines an equiregular sub-Riemannian structure and that f ⊂ ∆ s \ ∆ s − for some s ≥ . Also, assume that the complexities are measured w.r.t. the costfunction J ( u, T ) = k u k L ([0 ,T ] , R m ) or I ( u, T ) = k (1 , u ) k L ([0 ,T ] , R m +1 ) . We then have the following.i. Let Γ ⊂ M be a curve and define κ = max { k : T p Γ ∈ ∆ k ( p ) \ ∆ k − ( p ) , for any p in an opensubset of Γ } . Then, whenever the maximal time of definition of the controls T is sufficientlysmall, it holds Σ int (Γ , ε ) ≍ Σ app (Γ , ε ) ≍ ε κ . ii. On the other hand, let γ : [0 , T ] → M be a path such that f ( γ ( t )) = ˙ γ ( t ) mod ∆ s − ( γ ( t )) forany t ∈ [0 , T ] and define κ = max { k : γ ( t ) ∈ ∆ k ( γ ( t )) \ ∆ k − ( γ ( t )) for any t in an open subsetof [0 , T ] } . Then, it holds σ int ( γ, ε ) ≍ σ app ( γ, ε ) ≍ ε max { κ,s } , where the first asymptotic equivalence is true only when δ , i.e., the maximal time-step in σ int ( γ, ε ) , is sufficiently small. This theorem shows that, asymptotically, the complexity of curves is not influenced by the drift,and only depends on the underlying sub-Riemannian system, while the one of paths depends alsoon how “bad” the drift is with respect to this system. We remark also that for the neighboringapproximation complexity, σ app , it is not necessary to have an a priori bound on T .1.4. Long time local controllability.
As an application of the above theorem, let us brieflymention the problem of long time local controllability (henceforth simply
LTLC ), i.e., the problemof staying near some point for a long period of time
T >
0. This is essentially a stabilizationproblem around a non-equilibrium point.Since the system (1) satisfies the strong H¨ormander condition, it is always possible to satisfysome form of LTLC. Hence, it makes sense to quantify the minimal cost needed, by posing thefollowing. Let
T > q ∈ M , and γ q : [0 , T ] → M , γ q ( · ) ≡ q . • LTLC complexity by time:
LTLC time ( q , T, ε ) = σ int ( γ q , ε ) . Here, we require trajectories defined by admissible controls to pass through q at intervalsof time such that the minimal average cost between each passage is less than ε . • LTLC complexity by cost:
LTLC cost ( q , T, ε ) = σ app ( γ q , ε ) . Admissible controls for this complexity, will always be contained in the sub-Riemannianball of radius ε centered at q . learly, if f ( q ) = 0, then LTLC time ( q , T, δ ) = LTLC cost ( q , T, ε ) = 0, for any ε, δ, T > γ q is not a path by our definition, since it is not injective and ˙ γ q ≡
0, the arguments ofTheorem 1.3 can be applied also to this case. Hence, we get the following asymptotic estimate forthe LTLC complexities.
Corollary 1.4.
Assume that { f , . . . , f m } defines an equiregular sub-Riemannian structure andthat f ⊂ ∆ s \ ∆ s − for some s ≥ . Also, assume that the complexities are measured w.r.t. thecost function J ( u, T ) = k u k L ([0 ,T ] , R m ) or I ( u, T ) = k (1 , u ) k L ([0 ,T ] , R m +1 ) . Then, for any q ∈ M and T > it holds LTLC time ( q , T, δ ) ≍ LTLC cost ( q , T, ε ) ≍ ε s . Structure of the paper.
In Section 2 we introduce more in detail the setting of the prob-lem. In Section 3 we present some technical results regarding families of coordinates dependingcontinuously on the base point. These results will be essential in the sequel. Section 4 collects someuseful properties of the costs J and I , proved mainly in [25], while Section 5 is devoted to relatethe complexities of the control-affine system with those of the associated sub-Riemannian systems,and to prove Theorem 1.2. In this section we also prove Proposition 5.5, that gives a first resultin the direction of Theorem 1.3 showing when the sub-Riemannian and control-affine complexitiescoincide. Finally, the proof of the main result is contained in Sections 6 and 7, for curves and pathsrespectively. 2. Preliminaries
Throughout this paper, M is an n -dimensional connected smooth manifold.2.1. Sub-Riemannian control systems.
As already stated, a sub-Riemannian (or non-holonomic)control system on a connected smooth manifold M is a control system in the form(SR) ˙ q ( t ) = m X i =1 u i ( t ) f i ( q ( t )) , a.e. t ∈ [0 , T ]where u : [0 , T ] → R m is an integrable and bounded control function and { f , . . . , f m } is a familyof smooth vector fields on M . We let f u = P mi =1 u i f i . The value function d SR associated with theL cost is in fact a distance, called Carnot-Carath´eodory (or sub-Riemannian ) distance . Namely,for any q, q ′ ∈ M , d SR ( q, q ′ ) = inf Z T vuut m X j =1 u j ( t ) dt, where the infimum is taken among all the controls u ∈ L ([0 , T ] , R m ), for some T >
0, such thatits trajectory in (SR) is such that q u (0) = q and q u ( T ) = q ′ . An absolutely continuous curve γ : [0 , T ] → M is admissible for (SR) if there exists u ∈ L ([0 , T ] , R m ) such that ˙ γ ( t ) = f u ( t ).Let ∆ be the C ∞ -module generated by the vector fields { f , . . . , f m } (in particular, it is closedunder multiplication by C ∞ ( M ) functions and summation). Let ∆ = ∆, and define recursively∆ s +1 = ∆ s +[∆ s , ∆], for every s ∈ N . Due to the Jacobi identity ∆ s is the C ∞ -module of linearcombinations of all commutators of f , . . . , f m with length ≤ s . For q ∈ M , let ∆ s ( q ) = { f ( q ) : f ∈ ∆ s } ⊂ T q M . We say that { f , . . . , f m } satisfies the H¨ormander condition (or that it is a bracket-generating family of vector fields) if S s ≥ ∆ s ( q ) = T q M for any q ∈ M . Moreover, { f , . . . , f m } defines an equiregular sub-Riemannian structure if dim ∆ i ( q ) does not depend on the point for any i ∈ N . In the following we will always assume these two conditions to be satisfied.By the Chow–Rashevsky theorem (see for instance [1]), the hypothesis of connectedness of M and the H¨ormander condition guarantee the finiteness and continuity of d SR with respect to the opology of M . Hence, the sub-Riemannian distance, induces on M a metric space structure. Theopen balls of radius ε > q ∈ M , with respect to d SR , are denoted by B SR ( q, ε ).We say that a control u ∈ L ([0 , T ] , R m ), T >
0, is a minimizer of the sub-Riemannian distancebetween q, q ′ ∈ M if the associated trajectory q u with q u (0) = q is such that q u ( T ) = q ′ and k u k L ([0 ,T ] , R m ) = d SR ( q, q ′ ). Equivalently, u is a minimizer between q, q ′ ∈ M if it is a solution ofthe free-time optimal control problem, associated with (SR),(5) k u k L (0 ,T ) = Z T vuut m X j =1 u j ( t ) dt → min , q u (0) = q, q u ( T ) = q ′ , T > . It is a classical result that, for any couple of points q, q ′ ∈ M sufficiently close, there exists at leastone minimizer. Remark 2.1.
This control theoretical setting can be stated in purely geometric terms even ifwe drop the equiregularity assumption. Indeed, it is equivalent to a generalized sub-Riemannian structure. Such a structure is defined by a rank-varying smooth distribution and a Riemannianmetric on it (see [1] for a precise definition).In a sub-Riemannian control system, in fact, the map q span { f ( q ) , . . . , f m ( q ) } ⊂ T q M definesa rank-varying smooth distribution, which is naturally endowed with the Riemannian norm defined,for v ∈ ∆( q ), by(6) g ( q, v ) = inf (cid:26) | u | = q u + · · · + u m : f u ( q ) = v (cid:27) . The pair (∆ , g ) is thus a generalized sub-Riemannian structure on M . Conversely, every rank-varying distribution is finitely generated, see [2, 1, 3, 12], and thus a sub-Riemannian distance canbe written, globally, as the value function of a control system of the type (SR).Since { f , . . . , f m } is bracket-generating, the values of the sets ∆ s at q form a flag of subspacesof T q M , ∆ ( q ) ⊂ ∆ ( q ) ⊂ . . . ⊂ ∆ r ( q ) = T q M. The integer r , which is the minimum number of brackets required to recover the whole T q M , is called degree of non-holonomy (or step ) of the family { f , . . . , f m } at q . The degree of non-holonomy isindependent of q since we assumed the family { f , . . . , f m } to define an equiregular sub-Riemannianstructure. Let n s = dim ∆ s ( q ) for any q ∈ M . The integer list ( n , . . . , n r ) is called the growthvector associated with (SR). Finally, let w ≤ . . . ≤ w n be the weights associated with the flag,defined by w i = s if n s − < i ≤ n s , setting n = 0.For any smooth vector field f , we denote its action, as a derivation on smooth functions, by f : a ∈ C ∞ ( M ) f a ∈ C ∞ ( M ). For any smooth function a and every vector field f with f q , their (non-holonomic) order at q isord q ( a ) = min { s ∈ N : ∃ i , . . . , i s ∈ { , . . . , m } s.t. ( f i . . . f i s a )( q ) = 0 } , ord q ( f ) = max { σ ∈ Z : ord q ( f a ) ≥ σ + ord q ( a ) for any a ∈ C ∞ ( M ) } . In particular it can be proved that ord q ( a ) ≥ s if and only if a ( q ′ ) = O (d SR ( q ′ , q )) s . Definition 2.2. A system of privileged coordinates at q for { f , . . . , f m } is a system of local coor-dinates z = ( z , . . . , z n ) centered at q and such that ord q ( z i ) = w i , 1 ≤ i ≤ n .Let q ∈ M . A set of vector fields { f , . . . , f n } such that(7) { f ( q ) , . . . , f n ( q ) } is a basis of T q M, and f i ∈ ∆ w i for i = 1 , . . . , n, s called an adapted frame at q . We remark that to any system of privileged coordinates z at q isassociated a (non-unique) adapted frame at q such that ∂ z i = z ∗ f i ( q ) (i.e., privileged coordinatesare always linearly adapted to the flag).For any ordering { i , . . . , i n } , the inverse of the local diffeomorphisms( z , . . . , z n ) e z i f i + ··· + z in f in ( q ) , ( z , . . . , z n ) e z in f in ◦ · · · ◦ e z i f i ( q ) , defines privileged coordinates at q , called canonical coordinates of the first kind and of the secondkind , respectively. We remark that, for the canonical coordinates of the second kind, it holds z ∗ f i n ( z ) ≡ ∂ z in .We recall the celebrated Ball-Box Theorem, that gives a rough description of the shape of smallsub-Riemannian balls. Theorem 2.3 (Ball-Box Theorem) . Let z = ( z , . . . , z n ) be a system of privileged coordinates at q ∈ M for { f , . . . , f m } . Then there exist C, ε > such that for any ε < ε , it holds Box (cid:18) C ε (cid:19) ⊂ B SR ( q, ε ) ⊂ Box ( Cε ) , where, B SR ( q, ε ) is identified with its coordinate representation z (B SR ( q, ε )) and, for any η > , welet (8) Box ( η ) = { z ∈ R n : | z i | ≤ η w i } , Remark 2.4.
Let N ⊂ M be compact and let { z q } q ∈ N be a family of systems of privilegedcoordinates at q depending continuously on q . Then there exist uniform constants C, ε > q ∈ N in the system z q .2.2. Control-affine systems.
Let f and { f , . . . , f m } be smooth vector fields on M and, for some T >
0, define U T = S
0, steers from p to e − T f q the time-dependent control system(TD) ˙ q ( t ) = m X j =1 u j ( t ) ( e − tf ) ∗ f j ( q ( t )) . Sometimes proofs will be eased by considering (TD) instead of (D), due to the linearity w.r.t. thecontrol of the former.In the following we will often consider also the two sub-Riemannian control systems associatedwith (D), called respectively small and big, and defined as˙ q ( t ) = m X j =1 u j ( t ) f j ( q ( t )) , (SR-s) ˙ q ( t ) = u ( t ) f ( q ( t )) + m X j =1 u j ( t ) f j ( q ( t )) . (SR-b)We will denote by d SR and B SR the Carnot-Carath´eodory metric and metric balls, respectively,associated with (SR-s). This distance is well-defined due to Hypothesis (H2)3. Continuous families of coordinates
In this section we consider properties of families of coordinates depending continuously on thepoints of the curve or path, in order to be able to exploit Remark 2.4.From the definition of privileged coordinates, we immediately get the following.
Proposition 3.1.
Let γ : [0 , T ] → M be a path. Let t > and let z be a system of privilegedcoordinates at γ ( t ) for { f , . . . , f m } . Then, there exists C > such that (10) | z j ( γ ( t + ξ )) | ≤ C | ξ | for any j = 1 , . . . , n and any t + ξ ∈ [0 , T ] . Moreover, if for k ∈ N it holds that ˙ γ ( t ) / ∈ ∆ k − ( γ ( t )) , then there exist C , C , ξ > and acoordinate z α , of weight ≥ k , such that for any t ∈ [0 , T ] and any | ξ | ≤ ξ with t + ξ ∈ [0 , T ] it holds (11) C ξ ≤ z α ( γ ( t + ξ )) ≤ C ξ. Finally, if ˙ γ ( t ) ∈ ∆ k ( γ ( t )) \ ∆ k − ( γ ( t )) , the coordinate z α can be chosen to be of weight k .Proof. By the smoothness of γ , there exists a constant C > | ( z j ) ∗ ˙ γ ( t + ξ ) | ≤ C for any j = 1 , . . . , n and any t + ξ ∈ [0 , T ]. Thus, we obtain | z j ( γ ( t + ξ )) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t + ξt | ( z j ) ∗ ˙ γ ( t + η ) | dη (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ξ | . Let us prove (11). Let { f , . . . , f n } be an adapted basis associated with the system of coordinates z . In particular it holds that z ∗ f i ( γ ( t )) = ∂ z i . Moreover, let k ′ ≥ k be such that ˙ γ ( t ) ∈ ∆ k ′ ( γ ( t )) \ ∆ k ′ − ( γ ( t )) and write ˙ γ ( t ) = P w i ≤ k ′ a i ( t ) f i ( γ ( t )) for some a i ∈ C ∞ ([0 , T ]). Hence z ∗ ˙ γ ( t ) = X w i ≤ k ′ a i ( t ) z ∗ f i ( γ ( t )) = X w i ≤ k ′ a i ( t ) ∂ z i . Since there exists i with w i = k ′ such that a i ( t ) = 0, this implies that ( z i ) ∗ ˙ γ ( t ) = 0. Since k ′ ≥ k ,we have then proved (10). (cid:3) s already observed in Remark 2.4, in order to be apply the estimates of Theorem 2.3 uniformlyon γ we need to consider a continuous family of coordinates { z t } t ∈ [0 ,T ] such that each z t is privilegedat γ ( t ) for { f , . . . , f m } . We will call such a family a continuous coordinate family for γ .Let us remark that, fixed any basis { f , . . . , f n } adapted to the flag in a neighborhood of γ ([0 , T ]),letting z t be the inverse of the diffeomorphism(12) ( z , . . . , z n ) e z f ◦ . . . ◦ e z n f n ( γ ( t )) , defines a continuous coordinate family for γ .The following proposition precises Proposition 3.1. Proposition 3.2.
Let γ : [0 , T ] → M be a path and let k ∈ N such that ˙ γ ( s ) ∈ ∆ k ( γ ( s )) forany t ∈ [0 , T ] . Then, for any continuous coordinate family { z t } t ∈ [0 ,T ] for γ there exists constants C, ξ > such that for any t ∈ [0 , T ] and ≤ ξ ≤ ξ it holds (13) | z tj ( γ ( t + ξ )) | ≤ Cξ if w j ≤ k and | z tj ( γ ( t + ξ )) | ≤ Cξ wjk if w j > k. Proof.
Fix t ∈ [0 , T ] and let { f , . . . , f n } be an adapted basis associated with the privileged coordi-nate system z t . To lighten the notation, we do not explicitly write the dependence on time of suchbasis. Writing z t ∗ f i ( z ) = P nj =1 f ji ( z ) ∂ z tj , it holds that f ji is of weighted order ≥ w j − w i , and hencethere exists a constant C > | f ji ( z ) | ≤ C k z k ( w j − w i ) + . Here k z k is the pseudo-norm | z | w + · · · + | z n | w n and h + = max { , h } for any h ∈ R . Due to thecompactness of [0 , T ], the constant C can be choosen to be uniform w.r.t. the time.Since ˙ γ ( ξ ) ∈ ∆ k ( γ ( ξ )) for ξ >
0, there exist functions a i ∈ C ∞ ([0 , T ]) such that(15) ˙ γ ( ξ ) = X w i ≤ k a i ( ξ ) f i ( γ ( ξ )) for any ξ ∈ [0 , T ] . Observe that, for any t ∈ [0 , T ], it holds that(16) 1 ξ Z t + ξt | a i ( η ) | dη = | a i ( t ) | + O ( ξ ) as ξ ↓ , where O ( ξ ) is uniform w.r.t. t . In particular, for any ξ sufficiently small, this integral is bounded.By (15), for any t ∈ [0 , T ] we get(17) z tj ( γ ( t + ξ )) = X w i ≤ k Z t + ξt a i ( η ) f ji ( z t ( γ ( η ))) dη, for any t + ξ ∈ [0 , T ]Then, applying (14) we obtainmax ρ ∈ [0 ,ξ ] | z tj ( γ ( t + ρ )) | ≤ X w i ≤ k Z t + ξt | a i ( η ) | | f ji ( z t ( γ ( η ))) | dη ≤ C (cid:18) max ρ ∈ [0 ,ξ ] k z t ( γ ( t + ρ )) k (cid:19) ( w j − k ) + X w i ≤ k Z t + ξt | a i ( η ) | dη. (18) p to enlarging the constant C , this and (16) yieldmax ρ ∈ [0 ,ξ ] | z tj ( γ ( t + ρ )) | ξ w j ≤ C max ρ ∈ [0 ,ξ ] k z t ( γ ( t + ̺ k )) k ξ ! ( w j − k ) + X w i ≤ k ξ k Z t + ξ k t | a i ( η ) | dη ≤ C max ρ ∈ [0 ,ξ ] k z t ( γ ( t + ̺ k )) k ξ ! ( w j − k ) + . (19)Clearly, if max ρ ∈ [0 ,ξ ] k z t ( γ ( t + ρ k )) k /ξ ≤ C uniformly in t , inequality (19) proves (13). Then, letus assume by contradiction that max ρ ∈ [0 ,ξ ] k z t ( γ ( t + ρ k )) k /ξ is unbounded as ξ ↓
0. For any ξ let¯ ξ ∈ [0 , ξ ] to be such that k z t ( γ ( t + ¯ ξ k )) k = max ρ ∈ [0 ,ξ ] k z t ( γ ( t + ρ k )) k . Then, there exists a sequence ξ ν → + ∞ such that b ν = | z tj ( γ ( t + ¯ ξ kν )) | ξ w j ν −→ + ∞ and 1 n k z t ( γ ( t + ¯ ξ kν )) k ξ ν ≤ b wj ν ≤ k z t ( γ ( t + ¯ ξ kν )) k ξ ν . Moreover, by (19), it has to hold that w j > k . Then, again by (19), follows that b ν ≤ Cn b − kwj ν −→ ν → + ∞ . This contradicts the fact that b ν → + ∞ , and proves that there exists ξ >
0, a priori dependingon t , such that k z t ( γ ( t + ¯ ξ k )) k /ξ ≤ C for any ξ < ξ . Since [0 , T ] is compact, both constants ξ , C are uniform for t ∈ [0 , T ], thus completing the proof of (13) and of the proposition. (cid:3) We now focus on coordinate systems adapted to the drift. In particular, if for some s ∈ N itholds that f ⊂ ∆ s \ ∆ s − , it makes sense to consider the following definition. Definition 3.3. A privileged coordinate system adapted to f at q is a system of privileged coordi-nates z at q for { f , . . . , f m } such that there exists a coordinate z ℓ such that z ∗ f ≡ ∂ z ℓ .Observe that completing f to an adapted basis { f , . . . , f , . . . , f n } allows us to consider thecoordinate system adapted to f at q , given by the inverse of the diffeomorphism(20) ( z , . . . , z n ) e z ℓ f ◦ . . . ◦ e z n f n ( q ) . The following definition combines continuous coordinate families for a path γ : [0 , T ] → M withcoordinate systems adapted to a drift. Definition 3.4. A continuous coordinate family for γ adapted to f is a continuous coordinatefamily { z t } t ∈ [0 ,T ] for γ , such that each z t is a privileged coordinate system adapted to f at γ ( t ).Such coordinates systems can be built as per (20), letting the point q vary on the curve.Recall that f ⊂ ∆ s \ ∆ s − for some s , and consider a path γ : [0 , T ] → M such that ˙ γ ( t ) ∈ ∆ s ( γ ( t )) and that f ( γ ( t )) = ˙ γ ( t ) mod ∆ s − ( γ ( t )) for any t ∈ [0 , T ]. In this case, there exists f α ⊂ ∆ s \ ∆ s − and two functions ϕ ℓ , ϕ α ∈ C ∞ ([0 , T ]), ϕ α ≥
0, such that˙ γ ( t ) mod ∆ s − ( γ ( t )) = ϕ ℓ ( t ) f ( γ ( t )) + ϕ α ( t ) f α ( γ ( t )) . Moreover, by the assumption f ( γ ( t )) = ˙ γ ( t ) mod ∆ s − ( γ ( t )), if ϕ ℓ ( t ) = 1 then ϕ α ( t ) >
0. Then,using f α as an element of the adapted basis used to define a continuous coordinate family for γ adapted to f , it holds ( z ti ) ∗ ˙ γ ( t ) = ϕ i ( t ) for i = α, ℓ and any t ∈ [0 , T ]. The following lemma willbe essential to study this case. Lemma 3.5.
Assume that there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − . Let γ : [0 , T ] → M be apath such that ˙ γ ( t ) ∈ ∆ s ( γ ( t )) and such that f ( γ ( t )) = ˙ γ ( t ) mod ∆ s − ( γ ( t )) for any t ∈ [0 , T ] .Consider the continuous coordinate family { z t } t ∈ [0 ,T ] for γ adapted to f defined above. Then, there xist constants ξ , ρ, m > and a coordinate α = ℓ of weight s such that for any t ∈ [0 , T ] and ≤ ξ ≤ ξ , it holds ( z tℓ ) ∗ ˙ γ ( t + ξ ) ≤ − ρ if t ∈ E = { ϕ ℓ < − ρ } , (21) ( z tα ) ∗ ˙ γ ( t + ξ ) ≥ m if t ∈ E = { − ρ ≤ ϕ ℓ ≤ ρ } , (22) ( z tℓ ) ∗ ˙ γ ( t + ξ ) ≥ ρ if t ∈ E = { ϕ ℓ > ρ } . (23) In particular, it holds that E ∪ E ∪ E = [0 , T ] .Proof. Since ϕ α > ϕ − ℓ (1), by continuity of ϕ ℓ and ϕ α there exists ρ > ϕ α > ϕ − ℓ ([1 − ρ, ρ ]). Since E = ϕ − ℓ ([1 − ρ, ρ ]) is closed, letting 2 m = min E ϕ α > t, ξ ) ( z tα ) ∗ ˙ γ ( t + ξ ) on E × [0 , ξ ], for sufficiently small ξ . Finally, the uniform continuity of ( t, ξ ) ( z tℓ ) ∗ ˙ γ ( t + ξ ) over E × [0 , ξ ] and E × [0 , ξ ] yields(21) and (23). (cid:3) We end this section by observing that when the path is well-behaved with respect to the sub-Riemannian structure, it is possible to construct a very special continuous coordinate family, rec-tifying both γ and f at the same time. Proposition 3.6.
Let γ : [0 , T ] → M be a path and k ∈ N be such that ˙ γ ( t ) ∈ ∆ k ( γ ( t )) \ ∆ k − ( γ ( t )) for any t ∈ [0 , T ] , there exists a continuous coordinate family { z t } [0 ,T ] for γ adapted such that(1) there exists a coordinate z α of weight k such that z t ∗ ˙ γ ≡ ∂ z α ;(2) for any ξ, t ∈ [0 , T ] it holds that z tα = z t − ξα + ξ and z ti = z ξi if i = α .Moreover, if there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − and such that f ( γ ( t )) = ˙ γ ( t ) mod ∆ s − ( γ ( t )) for any t ∈ [0 , T ] whenever s = k , such family can be chosen adapted to f .Proof. By the assumptions on ˙ γ , it is possible to choose f α ⊂ ∆ k \ ∆ k − such that ˙ γ ( t ) = f α ( γ ( t )).Let then { f , . . . , f n } be the adapted basis obtained by completing f α and f . Finally, to completethe proof it is enough to consider the family of coordinates given by the inverse of the diffeomor-phisms ( z , . . . , z n ) e z ℓ f ◦ · · · ◦ e z α f α ( γ ( t )) . (cid:3) Cost functions
In this section we focus on properties of the cost functions defined in (3) and of the associatedvalue functions, respectively denoted by V J ( · , · ) and V I ( · , · ). For J such function is defined by(24) V J ( q, q ′ ) = inf (cid:8) J ( u, T ) | T > , q u (0) = q, q u ( T ) = q ′ (cid:9) . The definition of V I is analogous.4.1. Regularity of the value function.
The following result, in the case of J is contained in[25, Proposition 4.1], The proof can easily be extended to I . Theorem 4.1.
For any T > , the functions V J and V I are continuous from M × M → [0 , + ∞ ) (in particular they are finite). Moreover, for any q, q ′ ∈ M it holds V J ( q, q ′ ) ≤ min ≤ t ≤T d SR ( e tf q, q ′ ) , V I ( q, q ′ ) ≤ min ≤ t ≤T (cid:0) t + d SR ( e tf q, q ′ ) (cid:1) . Here e tf denotes the flow of f at time t and d SR denotes the Carnot-Carath´eodory distance w.r.t.the system (SR-s) , obtained from (D) by putting f = 0 . e remark that this fact follows from the following proposition (obtained adapting [25, Lemma3.6] to control-affine systems). Proposition 4.2.
For any η > sufficiently small and for any q , q ∈ M , it holds inf {J ( u, η ) | if q u (0) = q then q u ( η ) = q } ≤ d SR ( q , q ) . We denote the reachable set from the point q ∈ M with cost J less than ε > R f T ( q, ε ) = (cid:8) p ∈ M | V J ( q, p ) ≤ ε (cid:9) . Recall de definition of Box ( η ) in (8) and that { ∂ z i } ni =1 is the canonical basis in R n . Then, we definethe following sets, for parameters η > T > T ( η ) = [ ≤ ξ ≤ T (cid:18) ξ∂ z ℓ + Box ( η ) (cid:19) , Π T ( η ) = Box ( η ) ∪ [ <ξ ≤ T { z ∈ R n : 0 ≤ z ℓ − ξ ≤ η s , | z i | ≤ η w i + ηξ wis for w i ≤ s, i = ℓ, and | z i | ≤ η ( η + ξ s ) w i − for w i > s } . In [25] is proved a more general version of the following result, in the same spirit of Theorem 2.3.
Theorem 4.3.
Assume that there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − . Assume, moreover, that z = ( z , . . . , z n ) is a privileged coordinate system adapted to f , i.e., such that z ∗ f = ∂ z ℓ . Then,there exist C, ε , T > such that (26) Ξ T (cid:18) C ε (cid:19) ⊂ R f T ( q, ε ) ⊂ Π T ( Cε ) , for ε < ε and T < T . Here, with abuse of notation, we denoted by R f T ( q, ε ) the coordinate representation of the reachableset. In particular, (27) Box (cid:18) C ε (cid:19) ∩ { z ℓ ≤ } ⊂ R f T ( q, ε ) ∩ { z ℓ ≤ } ⊂ Box ( Cε ) ∩ { z ℓ ≤ } . Remark 4.4.
Let N ⊂ M be compact and let { z q } q ∈ N be a family of systems of privilegedcoordinates at q depending continuously on q . Then, as for Theorem 2.3 (see Remark 2.4), thereexist uniform constants C, ε , T > q ∈ N in the system z q .We notice also that, since [25, Example 21] is easily extendable to I , it follows that, for neither J nor I , the existence of minimizers is assured. Recall that a control u ∈ U T is a minimizer between q , q ∈ M for the cost J if its associated trajectory with initial condition q u (0) = q is such that q u ( T ) = q and V J ( q , q ) = J( u, T ).4.2. Behavior along the drift.
The following proposition assures that a minimizer for J and I always exists when moving in the drift direction. Proposition 4.5.
Assume that there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − . For any < t < T ,the unique minimizer between any q ∈ M and e tf q for the cost J is the null control on [0 , t ] .Moreover, if f / ∈ ∆( q ) , i.e. s ≥ , and the maximal time of definition of the controls T issufficiently small, the same is true for I .Proof. Since, for t ∈ [0 , T ], we have that V J ( q, e tf q ) = 0, the first statement is trivial.To prove the second part of the statement we proceed by contradiction. Namely, we assume thatthere exists a sequence T n −→ n ∈ N there exists a control v n ∈ L ([0 , t n ] , R m ) ⊂U T n , v n q to e T n f ( q ) and such that(28) t n + k v n k L ([0 ,t n ] , R m ) = I ( v n , t n ) ≤ I (0 , T n ) = T n . et z = ( z , . . . , z n ) be a privileged coordinate system adapted to f at q , as per Definition 3.3.Thus, by Theorem 4.3, it holds | z ℓ ( e T n f ( q )) | ≤ t n + C k v n k ([0 ,t n ] , R m ) . (29)Since z ℓ ( e T n f ( q )) = T n , putting together (28) and (29) yields k v n k L ([0 ,t n ] , R m ) ≤ C k v n k ([0 ,t n ] , R m ) for any n ∈ N . Since by the continuity of V I we have that k v n k L ([0 ,t n ] , R m ) →
0, this is a contra-diction. (cid:3)
We remark that, in the case of I , the assumption f / ∈ ∆( q ) of Proposition 4.5 is essential. Inparticular, in the following example we show that when f ⊂ ∆ even if a minimizer between q and e tf ( q ) exists, it could not coincide with an integral curve of the drift. Example 4.6.
Consider the control-affine system on R ,(30) ddt x = f ( x ) + u f ( x ) + u f ( x ) , where f = (1 ,
0) and f = ( φ , φ ) for some φ , φ : R → R , with φ = 0 and ∂ x ( φ /φ ) | (0 , = 0.Since f and f are always linearly independent, the underlying small sub-Riemannian system isindeed Riemannian with metric g = − φ /φ − φ /φ − φ φ ! . Let us now prove that the curve γ : [0 , → R , γ ( t ) = ( t T,
0) is not a minimizer of theRiemannian distance between (0 ,
0) and ( T, γ is not ageodesic for small T >
0. For γ the geodesic equation writes ( t Γ ( γ ( t )) = 0 ,t Γ ( γ ( t )) = 0 , for any t ∈ [0 , ⇐⇒ Γ ( · ,
0) = Γ ( · ,
0) = 0 near 0 . Here, Γ ikℓ are the Christoffel numbers of the second kind associated with g . A simple computationshows that Γ = φ φ ∂ x (cid:18) φ φ (cid:19) , Γ = ∂ x (cid:18) φ φ (cid:19) . Thus, if ∂ x ( φ /φ ) | (0 , = 0, then Γ (0 , = 0, showing that γ is not a geodesic.We now show that this fact implies that for any minimizing sequence u n = ( u n , u n ) ∈ L ([0 , t n ] , R for V I between (0 ,
0) and e T f ((0 , T, J ( u n +1 , t n +1 ) ≤ J ( u n , t n ), then u n = 0for sufficiently big n . To this aim, fix any t n →
0, let u n ( s ) = u ( s/t n ) and q n ( · ) be the trajectoryassociated with u n in system (30). Moreover, let v = ( v , ∈ L ([0 , S ] , R ) be the minimizerof I between (0 ,
0) and ( T,
0) in the system ˙ x = 1 + v . Since the trajectory of v is exactly γ ,by rescaling it holds length( γ ) = I ( v, S ). Then, by standard results in the theory of ordinarydifferential equations, it follows that q n ( t n ) → ( T,
0) and the fact that γ is not a Riemannianminimizing curve implies that k u n k L = k u k L < length( γ ) = I ( v, S ) . Hence, for sufficiently big n it holds that I ( u n , t n ) < I ( v, S ), proving the claim.As a consequence of Proposition 4.5, we get the following property for the complexities definedin the previous section with respect to the costs J and I . It generalizes to the control-affine settingthe trivial minimality of the sub-Riemannian complexity on the path Γ = { q } . orollary 4.7. Assume that there exists s ≥ such that f ⊂ ∆ s \ ∆ s − . Let x ∈ M and y = e T f x , for some < T < T . Then, for any ε > , the minimum over all curves Γ ⊂ M (resp.paths γ : [0 , T ] → M ) connecting x and y of Σ J int ( · , ε ) and Σ J app ( · , ε ) (resp. σ J int ( · , δ ) and σ J app ( · , ε ) )is attained at Γ = { e tf } t ∈ [0 ,T ] (resp. at γ ( t ) = e tf x ). Moreover, the same is true for the cost I ,whenever T is sufficiently small. Behavior transversally to the drift.
When we consider two points on different integralcurves of the drift, it turns out that the two costs J and I are indeed equivalent, as proved in thefollowing. Proposition 4.8.
Assume that there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − . Let q, q ′ ∈ M be suchthat there exists a set of privileged coordinates adapted to f at q . Then, there exists C, ε , T > such that, for any u ∈ U T such that, for some T < T , q u ( T ) = q ′ and J ( u, T ) < ε , it holds J ( u, T ) ≤ I ( u, T ) ≤ C J ( u, T ) . The proof of this fact relies on the following particular case of [25, Lemma 25].
Lemma 4.9.
Assume that there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − . Let q ∈ M and let z = ( z , . . . , z n ) be a system of privileged coordinate system adapted to f at q . Then, there exist C, ε , T > such that, for any u ∈ U T , with J ( u, T ) < ε for some T < T , it holds T ≤ C (cid:0) J ( u, T ) s + z ℓ ( q u ( T )) + (cid:1) . Here, we let ξ + = max { ξ, } . This Lemma is crucial, since it allows to bound the time of definition of any control through itscost. We now prove Proposition 4.8.
Proof of Proposition 4.8.
The first inequality is trivial. The second one follows by applying Lemma 4.9,and computing I ( u, T ) ≤ T + J ( u, T ) ≤ ( Cε s − + 1) J ( u, T ) . (cid:3) First results on complexities
In this section we collect some first results regarding the various complexities we defined.Firstly, we prove a result on the behavior of complexities. For all the complexities under consid-reation, except the interpolation by time complexity, such result will hold with respect to a genericcost function J : U T → [0 , + ∞ ), satisfying some weak hypotheses. Proposition 5.1.
Assume that for any q ∈ M and any q / ∈ { e tf q } t ∈ [0 , T ] , it holds V J ( q , q ) > .Then, the following holds.i. For any curve Γ ⊂ M it holds the following.(a) If the maximal time of definition of the controls, T , is sufficiently small, then lim ε ↓ Σ int (Γ , ε ) =lim ε ↓ Σ app (Γ , ε ) = + ∞ . (b) If Γ is an admissible curve for (D) , then ε Σ int (Γ , ε ) and ε Σ app (Γ , ε ) are bounded fromabove, for any ε > .ii. For any path γ : [0 , T ] → M it holds the following.(a) If γ is not a solution of (D) , lim ε ↓ σ app ( γ, ε ) = + ∞ .(b) If the cost is either J or I , f ⊂ ∆ s \ ∆ s − , ˙ γ ( t ) ⊂ ∆ k ( γ ( t )) \ ∆ k − ( γ ( t )) and f ( γ ( t )) =˙ γ ( t ) mod ∆ s − ( γ ( t )) for any t ∈ [0 , T ] , then lim ε ↓ σ int ( γ, ε ) = + ∞ whenever δ < η .(c) If γ is an admissible curve for (D) , then εσ int ( γ, ε ) and ε σ app ( γ, ε ) are bounded by above,for any δ, ε > . ( · ) x y e j TN f ( x ) e · f ( x ) f Figure 5.
An example of a curve satisfying Remark 5.3, with a rectified drift.
Proof.
The last statement for curves and paths follows simply by considering the control whosetrajectory is the curve or the path itself, which is always admissible regardless of ε .We now prove the first statement for the interpolation by cost complexity of a curve Γ. Thesame reasonings will hold for Σ app and σ app . Let x, y be the two endpoints of Γ and assume T tobe sufficiently small so that V J ( x, y ) >
0. Then, the first statement follows fromlim ε ↓ Σ int (Γ , ε ) ≥ V ( x, y ) lim ε ↓ ε = + ∞ . Consider now the interpolation by time complexity and proceed by contradiction. Namely, letus assume that there exists a constant
C > σ int ( γ, ε ) ≤ C for any ε >
0. Then,by definition of σ int , this implies that for any ε > δ ε ∈ [ CT / , δ ) and a δ ε -timeinterpolation u ε ∈ L ([0 , T ] , R m ) such that δ ε J( u ε , T ) ≤ ε .Firstly, observe that by Lemma 4.9 and the assumptions on f and ˙ γ , we obtain that there exist η > I ⊂ [0 , T ], with | I | > η , such that(31) V (cid:0) γ ( t h ) , γ ( t h ) (cid:1) −→ h ↓ ⇒ t h − t h −→ h ↓ , whenever t h ∈ I and t h > t h for any h in a right neighborhood of zero.For any ε , let 0 = t ε < t ε < . . . < t εN ε = T be a partition of [0 , T ] such that q u ε ( t εi ) = γ ( t εi )for any i ∈ { , N ε } and t εi − t εi − ≤ δ ε . It is clear that, up to removing some t εi ’s, we can assumethat t εi − t εi − ≥ δ ε / ≥ CT /
4. Let us fix, τ ε = t εi ε ∈ I for some index i ε and τ ε = t εi ε +1 . Such τ ε always exists, since | I | > δ . Since, by the definition of u ε and the choice of the cost, follows thatV J ( γ ( τ ε ) , γ ( τ ε )) → ε ↓ ε ↓ (cid:0) τ ε − τ ε (cid:1) ≥ CT > . (cid:3) Remark 5.2.
Result ii.b, regarding the interpolation by time complexity, holds for any cost satis-fying the assumptions of Proposition 5.1, such that for any path γ it holds (31), and that, for any u ∈ L ([0 , T ] , R m ), there exists a constant such that, if t , t ∈ [0 , T ], t < t , thenJ( u | [ t ,t ] ( · + t ) , t − t ) ≤ C J( u, T ) . Remark 5.3.
The bound on δ in Proposition 5.1 is essential. For example, consider the cost J ( u, T ) = k u k L ([0 ,T ] , R m ) , and a curve such that, for some N ∈ N , it holds γ ( jT /N ) = e j ( T/N ) f ( γ (0))for any j = 1 , . . . , T /N (see, e.g., Figure 5). In this case, the null control is a ( T /N )-time interpo-lation of γ , with J (0 , T ) = 0. In particular, if δ > T /N , it holds σ int ( γ, ε ) ≤ N . n the following, we will denote with an apex “SR-s” – e.g. Σ SR-sint – the complexities associatedwith the small sub-Riemannian system (SR-s) defined at p. 10, and with an apex “SR-b”, e.g.Σ
SR-bint , the ones associated with the big sub-Riemannian system (SR-b).We immediately get the following.
Proposition 5.4.
Let Γ ⊂ M be a curve and γ : [0 , T ] → M be a path.(i) Any complexity relative to the cost J is smaller than the same complexity relative to I .Namely, for any ε, δ > , it holds Σ J int (Γ , ε ) ≤ Σ I int (Γ , ε ) , Σ J app (Γ , ε ) ≤ Σ I app (Γ , ε ) ,σ J int ( γ, ε ) ≤ σ I int ( γ, ε ) , σ J app ( γ, ε ) ≤ σ I app ( γ, ε ) . (ii) For any cost, the neighboring approximation complexity of some path is always bigger thanthe tubular approximation complexity of its support. Namely, for any γ : [0 , T ] → M andany ε > , it holds Σ J app ( γ ([0 , T ]) , ε ) ≤ σ app J ( γ, ε ) , Σ I app ( γ ([0 , T ]) , ε ) ≤ σ I app ( γ, ε ) (iii) Any complexity relative to the cost I is bigger than the same complexity computed for thesystem (SR-b) . Namely, for any ε, δ > , it holds Σ SR-b int (Γ , ε ) ≤ Σ I int (Γ , ε ) , Σ SR-b app (Γ , ε ) ≤ Σ I app (Γ , ε ) ,σ SR-b int ( γ, ε ) ≤ σ I int ( γ, ε ) , σ SR-b app ( γ, ε ) ≤ σ I app ( γ, ε ) . (iv) In the case of curves, the complexities relative to the cost I are always smaller than thesame complexities computed for the system (SR-s) . Namely, for any ε > it holds Σ I int (Γ , ε ) ≤ Σ SR-s int (Γ , ε ) , Σ I app (Γ , ε ) ≤ Σ SR-s app (Γ , ε ) . Proof.
The inequality in ( ii ) is immediate, since any control admissible for the σ app ( γ, ε ) is alsoadmissible for Σ app ( γ ([0 , T ]) , ε ).On the other hand, the inequalities in ( iii ) between the complexities in (SR-b) and the onesin (D), with cost I , is a consequence of the fact that, for every control u ∈ U T , the trajectory q u is admissible for (SR-b) and associated with the control u = (1 , u ) : [0 , T ] → R m +1 with k u k L ([0 ,T ] , R m +1 ) = I ( u, T ). The inequalities in ( i ) between the complexities in (D) with respectto the different costs follows from the fact that J ≤ I .Finally, to complete the proof of the proposition, observe that, by Theorem 4.1, it holds thatV I ( q, q ′ ) ≤ d SR ( q, q ′ ) , for any q, q ′ ∈ M. This shows, in particular, that every ε -cost interpolation for (SR-s), is an ε -cost interpolation for(D), proving the statement regarding the cost interpolation complexity in ( iv ). The part concerningthe tubular approximation follows in the same way. (cid:3) We conclude this section by proving an asymptotic equivalence for the complexities of a control-affine system in a very special case. In particular, we will prove that if we cannot generate thedirection of Γ with an iterated bracket of f and some f , . . . , f m , then the curve complexities forthe systems (D), (SR-s) and (SR-b) behaves in the same way.Let L f be the ideal of the Lie algebra Lie( f , f , . . . , f m ) generated by the adjoint endomorphismad( f ) : f ad( f ) f = [ f , f ], f ∈ Vec( M ). Then the following holds. Proposition 5.5.
Assume that there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − , and let Γ ⊂ M be acurve such that there exists k ∈ N for which T Γ ⊂ ∆ k \ ∆ k − . Assume, moreover, that for any q ∈ Γ it holds that T q Γ
6⊂ L f ( q ) . Then, for sufficiently small T , (32) Σ J int (Γ , ε ) ≍ Σ I int (Γ , ε ) ≍ Σ J app (Γ , ε ) ≍ Σ I app (Γ , ε ) ≍ ε k . roof. By the fact that T q Γ
6⊂ L f ( q ), follows that T q Γ ⊂ Lie kq ( f , f , . . . , f m ) \ Lie k − q ( f , f , . . . , f m ).Thus, approximating Γ in the big or in the small sub-Riemannian system is equivalent, and by The-orem 1.1 follows Σ SR-sint (Γ , ε ) ≍ Σ SR-bint (Γ , ε ) ≍ Σ SR-sapp (Γ , ε ) ≍ Σ SR-bapp (Γ , ε ) ≍ ε k . The statement then follows by applying Proposition 5.4. (cid:3)
Remark 5.6.
Observe that if f ∈ ∆ in a neighborhood U of Γ, it holds that Lie kq ( f , f , . . . , f m ) =∆ k ( q ) for any q ∈ U . Then, by the same argument as above, we get that (32) holds. This showsthat, where f ⊂ ∆, the asymptotic behavior of complexities of curves is the same as in thesub-Riemannian case. 6. Complexity of curves
This section is devoted to prove the statement on curves of Theorem 1.3. Namely, we will provethe following.
Theorem 6.1.
Assume that there exists s ≥ such that f ⊂ ∆ s \ ∆ s − . Let Γ ⊂ M be a curveand define κ = max { k : T p Γ ∈ ∆ k ( p ) \ ∆ k − ( p ) for some p ∈ Γ } . Then, if the maximal time ofdefinition of the controls T is small enough, Σ J int (Γ , ε ) ≍ Σ I int (Γ , ε ) ≍ Σ J app (Γ , ε ) ≍ Σ I app (Γ , ε ) ≍ ε κ , Due to the fact that the value functions associated with the costs J and I are always smallerthan the sub-Riemannian distance associated with system (SR-s), the immediately follows fromthe results in [23]. Proposition 6.2.
Let Γ ⊂ M be a curve such that there exists k ∈ N for which T Γ ⊂ ∆ k . Then, Σ J int (Γ , ε ) Σ I int (Γ , ε ) ε k , Σ J app (Γ , ε ) Σ I app (Γ , ε ) ε k . Proof.
By ( i ) in Proposition 5.4, follows that we only have to prove the upper bound for thecomplexities relative to the cost I . Moreover, by the same proposition and [23, Theorem 3.14],follows immediately that Σ I int (Γ , ε ) and Σ I app (Γ , ε ) ε − k , completing the proof of the proposition. (cid:3) In order to prove < , we will need to exploit a sub-additivity property of the complexities. In orderto have this property, it is necessary to exclude certain bad behaving points, called cusps. Nearthese points, the value function behaves like the Euclidean distance does near algebraic cusps (e.g.,(0 ,
0) for the curve y = p | x | in R ). In the sub-Riemannian context, they have been introduced in[23]. Definition 6.3.
The point q ∈ Γ is a cusp for the cost J if it is not an endpoint of Γ and if,for every c, η >
0, there exist two points q , q ∈ Γ such that q lies between q and q , with q before q and q after q w.r.t. the orientation of Γ (in particular q = q , q ), V J ( q , q ) ≤ η andV J ( q, q ) ≥ c V( q , q ).In [23] is proved that no curve has cusps in an equiregular sub-Riemannian stucture. As thefollowing example shows, the equiregularity alone is not enough for control-affine systems. Example 6.4.
Consider the following vector fields on R , with coordinates ( x, y, z ), f ( x, y, z ) = ∂ x , f ( x, y, z ) = ∂ y + x∂ z . Since [ f , f ] = ∂ z , { f , f } is a bracket-generating family of vector fields. The sub-Riemanniancontrol system associated with { f , f } on R corresponds to the Heisenberg group. et now f = ∂ z ⊂ ∆ \ ∆ be the drift, and let us consider the curve Γ = { ( t , , t ) | t ∈ ( − η, η ) } .Let q = (0 , , T q Γ / ∈ ∆( q ), by smoothness of Γ and ∆, for η sufficiently small T Γ ⊂ ∆ \ ∆.We now show that the point q is indeed a cusp for the cost J . In fact, for any ξ > ξ < T , it holds that the null control defined over time [0 , ξ ] steers the control affine system from q = ( ξ , , − ξ ) ∈ Γ to q = ( ξ , , ξ ) ∈ Γ. Hence, by Proposition 4.5, V J ( q , q ) = 0. Moreover,since q and q are not on the same integral curve of the drift, V J ( q, q ) > V J ( q , q ). Thisproves that q is a cusp for J .The following proposition shows that cusps appear only where the drift becomes tangent to thecurve at isolated points, as in the above example. Proposition 6.5.
Assume that there exists s ≥ such that f ⊂ ∆ s \ ∆ s − . Let Γ ⊂ M be a curvesuch that T Γ ⊂ ∆ k \ ∆ k − . Moreover, if s = k , let Γ be such that either f ( p ) / ∈ T p Γ ⊕ ∆ s − ( p ) forany p ∈ Γ or f | Γ ⊂ T Γ ⊕ ∆ s − . Then Γ has no cusps for the cost V J .Proof. If f | Γ ⊂ T p Γ ⊕ ∆ s − ( p ), the statement is a consequence of Proposition 4.5. Hence, weassume that f ( p ) / ∈ T p Γ ⊕ ∆ s − ( p ) for any p ∈ Γ. Let γ : [0 , T ] → M be a path parametrizing Γand consider the continuous coordinate family { z t } t ∈ [0 , T ] adapted to f given by Proposition 3.6.In particular, it holds that z t ∗ ˙ γ ( · ) ≡ ∂ z α for some coordinate z α of weight k and for any t ∈ [0 , T ].We now fix any t ∈ (0 , T ) and prove that γ ( t ) is not a cusp. In fact, letting η > z tℓ ( γ ( · )) ≡ V J ( γ ( t ) , γ ( t + η )) ≤ C n X j =1 | z t j ( γ ( t + η )) | wj = C | z t α γ ( t + η ) | k = 2 C | z t − ηα ( γ ( t + η )) | k ≤ CV ( γ ( t − η ) , γ ( t + η )) . Letting t = t − η and t = t + η , this proves that V J ( γ ( t ) , γ ( t )) ≤ V J ( γ ( t ) , γ ( t )). Bydefinition, this implies that γ ( t ) is not a cusp, completing the proof of the proposition. (cid:3) Finally, we can prove the sub-additivity of the curve complexities.
Proposition 6.6.
Let Γ ′ ⊂ Γ ⊂ M be two curves. Then, if the endpoints of Γ ′ are not cusps forthe cost J , there exists a constant C > such that for sufficiently small T it holds Σ J int (Γ ′ , ε ) Σ J int (Γ , ε ) , Σ J app (Γ ′ , ε ) Σ J app (Γ , ε ) . Proof. Cost interpolation complexity.
Let u ∈ L ([0 , T ] , R m ) be a control admissible for Σ J int (Γ , ε ),and let 0 = t < . . . < t N = T be such that k u k L ([ t i − ,t i ]) ≤ ε . Recall that by Theorem 4.1, V J isa continuous function. Since for small T >
0, for any ε > q ∈ M the reachable set R T ( q, ε ) is bounded, it holds that R T ( q, ε ) ց { e tf ( q ) | t ∈ [0 , T ] } as ε ↓
0, in the sense of pointwiseconvergence of characteristic functions. From this follows that, for ε and T sufficiently small, thereexist i = i such that q u ( t i ) ∈ Γ ′ for any i ∈ { i , . . . , i } and q u ( t i ) Γ ′ for any i / ∈ { i , . . . , i } .Since x ′ and y ′ are not cusps, there exists c > x ′ and y ′ be the endpoints of Γ ′ , itholds V J ( x ′ , q u ( t i )) ≤ c V I ( q u ( t i − , q u ( t i )) ≤ ε and V J ( q u ( t i ) , y ′ ) ≤ V J ( q u ( t i ) , q u ( t i +1 )) ≤ cε .Thus, there exists a constant C > J int (Γ ′ , ε ) ≤ J ( u | [ t i ,t i ] ) ε + 2 c ≤ C J ( u | [ t i − ,t i ] ) ε ≤ C J ( u ) ε . Taking the infimum over all controls u , admissible for Σ J int (Γ , ε ) completes the proof. Tubular approximation complexity.
Let u ∈ L ([0 , T ] , R m ) be a control admissible for Σ J app (Γ , ε ).Then, letting q u be its trajectory such that q u (0) = x , there exists two times t and t such that q u ( t ) ∈ B SR ( x ′ , Cε ) and q u ( t ) ∈ B SR ( y ′ , Cε ). Then, since V J ≤ d SR by Theorem 4.1, the sameargument as above applies. (cid:3) hanks to the sub-additivity, we can prove the < part of Theorem 6.1 in the case where thecurve is always tangent to the same stratum ∆ k \ ∆ k − . Proposition 6.7.
Assume, that there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − . Let Γ ⊂ M be acurve such that there exists k ∈ N for which T p Γ ∈ ∆ k ( p ) \ ∆ k − ( p ) for any p ∈ Γ . Then, forsufficiently small time T , it holds Σ I int (Γ , ε ) < Σ J int (Γ , ε ) < ε k , Σ I app (Γ , ε ) < Σ J app (Γ , ε ) < ε k . Proof.
By Proposition 5.4, Σ I int (Γ , ε ) < Σ J int (Γ , ε ) and Σ I app (Γ , ε ) < Σ J app (Γ , ε ). We will only provethat Σ J int (Γ , ε ) < ε − k , since the same arguments apply to Σ J app (Γ , ε ).Let γ : [0 , T ] → M be a path parametrizing Γ. We will distinguish three cases. Case 1 f ( p ) / ∈ ∆ s − ( p ) ⊕ T p Γ for any p ∈ Γ : Fix η > u ∈ L ([0 , T ] , R m ),admissible for Σ int (Γ , ε ) such that(33) k u k L ε ≤ Σ int (Γ , ε ) + η. Let u i = u | [ t i − ,t i ] , i = 1 , . . . , N = l k u k L1 ε m to be such that k u i k L = ε for any 1 ≤ i < N , k u N k L ≤ ε . Moreover, let s i be the times such that γ ( s i ) = q u ( t i ).By (33), it holds N ≤ ⌈ Σ int (Γ , ε ) + η + 1 ⌉ . However, we can assume w.l.o.g. that N ≤⌈ Σ int (Γ , ε ) + η ⌉ . In fact, N > ⌈ Σ int (Γ , ε ) + η ⌉ only if k u N k < ε . In this case we can simplyrestrict ourselves to compute Σ int (˜Γ , ε ) where ˜Γ is the segment of Γ comprised between x and q u ( t N − ). Indeed, by Propositions 6.5 and 6.6, it follows that Σ int (˜Γ , ε ) Σ int (Γ , ε ).We now assume that ε and T are sufficiently small, in order to satisfy the hypothesesof Theorem 4.3 at any point of Γ. Moreover, let { z t } t ∈ [0 , T ] be the continuous coordinatefamily for Γ adapted to f given by Proposition 3.6. Then, it holds(34) T = N X i =1 ( s i − s i − ) = N X i =1 | z s i − α ( γ ( s i )) | = N X i =1 | z s i − α ( q u ( t i )) | ≤ C (Σ int (Γ , ε ) + η ) ε k . Here, in the last inequality we applied Theorem 4.3 and the fact that z s i − ℓ ( q u ( t i )) = 0 byProposition 3.6. Finally, letting η ↓ ε sufficiently small itholds Σ int (Γ , ε ) ≥ C T ε − k . This completes the proof in this case. Case 2 s = k and f ( p ) ∈ ∆ s − ( p ) ⊕ T p Γ for any p ∈ Γ : Let { z t } t ∈ [0 , T ] be a continuous co-ordinate family for γ adapted to f . In this case, since ( z tℓ ) ∗ f = 1, it holds that ( z tℓ ) ∗ ˙ γ ( · ) =0. Hence, there exist C , C > t, ξ ∈ [0 , T ] C ( t − ξ ) ≤ z tℓ ( γ ( ξ )) ≤ C ( t − ξ ) , if ( z tℓ ) ∗ ˙ γ ( · ) > C ( t − ξ ) ≤ − z tℓ ( γ ( ξ )) ≤ C ( t − ξ ) , if ( z tℓ ) ∗ ˙ γ ( · ) < . (36) If (36) holds, then we can proceed as in Case 1 with α = ℓ . In fact, | z s i − ℓ ( q u ( t i ) | ≤ Cε s byTheorem 4.3. On the other hand, if (35) holds, by applying Theorem 4.3 we get T = N X i =1 ( s i − s i − ) ≤ C N X i =1 | z s i − ℓ ( γ ( s i )) | = 1 C N X i =1 | z s i − ℓ ( q u ( t i )) |≤ C N X i =1 ( Cε s + t i − t i − ) ≤ C (cid:0) Σ J int (Γ , ε ) + η (cid:1) ε s + T. By taking T sufficiently small, it holds T ≤ T < T . Then, letting η ↓ J int (Γ , ε ) ≥ (( T − T ) /C ) ε − s < ε − s . This completes the proof of this case. ase 3 s = k and f ( p ) ∈ ∆ s − ( p ) ⊕ T p Γ for some p ∈ Γ : In this case, there exists an openinterval ( t , t ) ⊂ [0 , T ] such that f ( γ ( t )) = ˙ γ ( t ) mod ∆ s − ( γ ( t )) for any t ∈ ( t , t ). Thus,Γ ′ = γ (( t , t )), satisfies the assumption of Case 1 and hence Σ J int (Γ ′ , ε ) < ε − k . Moreover, byProposition 6.5, we can assume that γ ( t ) and γ ( t ) are not cusps. Then, by Proposition 6.6we get 1 ε k Σ J int (Γ ′ , ε ) Σ J int (Γ , ε ) , completing the proof of the proposition. (cid:3) Finally, we are in a condition to prove the main theorem of this section.
Proof of Theorem 6.1.
Since it is clear that T Γ ⊂ ∆ κ , the upper bound follows by Proposition 6.2.Moreover, by Proposition 5.4 it suffices to prove that Σ J int (Γ , ε ) and Σ J app (Γ , ε ) < ε − κ . Since thearguments are analogous, we only prove this for Σ J int .By smoothness of Γ, the set A = { p ∈ Γ | T p Γ ∈ ∆ κ ( p ) \ ∆ κ − ( p ) } has non-empty interior. Letthen Γ ′ ⊂ A be a non-trivial curve such that either f ( p ) / ∈ T p Γ ′ ⊕ ∆ s − ( p ) for any p ∈ Γ ′ or that f | Γ ′ ⊂ T Γ ′ ⊕ ∆ s − . Then, since by Proposition 6.5 we can choose Γ ′ such that it does not containany cusps, applying Proposition 6.6 yields that Σ J int (Γ ′ , ε ) Σ J int (Γ , ε ). Finally, the result followsfrom the fact that, by Proposition 6.7, it holds Σ J int (Γ ′ , ε ) < ε − κ . (cid:3) Complexity of paths
In this section we will prove the statement on paths of Theorems 1.2 and 1.3.Recall the definition of δ -time interpolation given in Section 1.2, and define the following functionof a path γ : [0 , T ] → M and a time-step δ > ω ( γ, δ ) = δ inf (cid:8) J ( u, T ) | u is a δ -time interpolation of γ (cid:9) . Controls admissible for the above infimum define trajectories touching γ at intervals of time oflength at most δ . Then, function ω ( γ, δ ) measures the minimal average cost on each of theseintervals. It is possible to express the interpolation by time complexity through ω . Namely,(37) σ int ( γ, ε ) = inf δ ≤ δ (cid:26) Tδ (cid:12)(cid:12)(cid:12)(cid:12) ω ( γ, δ ) ≤ ε (cid:27) = sup δ ≤ δ (cid:26) Tδ (cid:12)(cid:12)(cid:12)(cid:12) ω ( γ, δ ′ ) ≥ ε for any δ ′ ≥ δ (cid:27) . From (37) follows immediately that, for any k ∈ N ,(38) σ int ( γ, ε ) ε − k ⇐⇒ ω ( γ, δ ) δ k and σ int ( γ, ε ) < ε − k ⇐⇒ ω ( γ, δ ) < δ k . Exploiting this fact, we are able to prove Theorem 1.2.
Proof of Theorem 1.2.
Let { z t } t ∈ [0 ,T ] to be the continuous family of coordinates for γ given byProposition 3.6. We start by proving that ω ( γ, δ ) δ k which, by (38), will imply σ SR-sint ( γ, ε ) ε − k . Fix any partition 0 = t < t < . . . < t N = T such that δ/ ≤ t i − t i − ≤ δ . If δ issufficiently small, from Theorem 2.3 follows that there exists a constant C > i = 0 , . . . , N in the coordinate system z t i it holds that Box( γ ( t i ) , Cδ k ) ⊂ B SR ( γ ( t i ) , δ k ). Hence,since z t i − α ( γ ( t i )) = t i − t i − , that z t i − j ( γ ( t i )) = 0 for any j = α , and that N ≤ ⌈ T /δ ⌉ ≤
CT /δ ,we get ω ( γ, δ ) ≤ δ N X i =1 d SR ( γ ( t i − ) , γ ( t i )) ≤ Cδ N X i =1 n X j =1 | z t i − j ( γ ( t i )) | wj = Cδ N X i =1 ( t i − t i − ) k ≤ CT δ k . This proves completes the proof of the first part of the Theorem. onversely, to prove that σ int ( γ, ε ) ε − k we need to show that ω ( γ, δ ) < δ k . To this aim, let η > u ∈ L be a control admissible for ω ( γ, δ ) such that k u k L ([ t i − ,t i ]) ≤ ω ( γ, δ ) δ + η. Let 0 = t < t < . . . < t N = T be times such that q u ( t i ) = γ ( t i ), i = 0 , . . . , N , 0 < t i − t i − ≤ δ .Moreover, let u i ∈ L ([ t i − , t i ]) be the restriction of u between t i − and t i . Observe that, up toremoving some t i ’s, we can assume that t i − t i − ∈ (cid:0) δ , δ (cid:3) . This implies that ⌈ T / (3 δ ) ⌉ ≤ N ≤⌈ T /δ ⌉ .To complete the proof it suffices to show that k u i k L ([ t i − ,t i ]) ≥ Cδ k . In fact, for any η >
0, thisyields ω ( γ, δ ) δ ≥ k u k L ([0 ,T ] , R m ) − η = N X i =1 k u i k L ([ t i − ,t i ]) − η ≥ C N X i =1 δ k − η ≥ C T δ δ k − η. Letting η ↓
0, this will prove that ω ( γ, δ ) < δ k , completing the proof.Observe that, by Theorem 2.3, for any i = 1 , . . . , N in the coordinate system z t i − it holdsB SR ( γ ( t i ) , k u i k L ([ t i − ,t i ]) ) ⊂ Box (cid:16) γ ( t i ) , C k u i k L ([ t i − ,t i ]) (cid:17) . Since z t i − α ( t i ) = t i − t i − , this impliesthat δ ≤ t i − t i − = | z t i − α ( γ ( t i )) | ≤ C k u i k k L ([ t i − ,t i ]) , proving the claim and the theorem. (cid:3) The rest of the section will be devoted to the proof of the statement on paths of Theorem 1.3.Namely, we will prove the following.
Theorem 7.1.
Assume that there exists s ≥ such that f ⊂ ∆ s \ ∆ s − . let γ : [0 , T ] → M be apath such that f ( γ ( t )) = ˙ γ ( t ) mod ∆ s − ( γ ( t )) for any t ∈ [0 , T ] and define κ = max { k : γ ( t ) ∈ ∆ k ( γ ( t )) \ ∆ k − ( γ ( t )) for any t in an open subset of [0 , T ] } . Then, it holds σ J int ( γ, ε ) ≍ σ I int ( γ, ε ) ≍ σ J app ( γ, ε ) ≍ σ I app ( γ, ε ) ≍ ε max { κ,s } , where the asymptotic equivalences regarding the interpolation by time complexity are true only when δ , i.e., the maximal time-step in σ int ( γ, ε ) , is sufficiently small. Differently to what happened for curves, the part does not immediately follow from theestimates of sub-Riemannian complexities, but requires additional care. It is contained in thefollowing proposition. Proposition 7.2.
Assume that there exists s ∈ N such that f ⊂ ∆ s \ ∆ s − . Let γ : [0 , T ] → M bea path such that ˙ γ ( t ) ∈ ∆ k ( γ ( t )) . Then, it holds (39) σ J int ( γ, ε ) σ I int ( γ, ε ) ε max { s,k } , σ J app (Γ , ε ) σ I app (Γ , ε ) ε max { s,k } . Proof.
By ( i ) in Proposition 5.4, follows that we only have to prove the upper bound for thecomplexities relative to the cost I . We will start by proving (39) for σ I int . In particular, by (38) itwill suffices to prove ω I ( γ, δ ) δ k Let { z t } t ∈ [0 ,T ] be a continuous coordinate family for γ adapted to f . Let ˜ γ t ( ξ ) = e − ( ξ − t ) f ( γ ( ξ )).Then, since z t ∗ f = ∂ z ℓ , it holds(40) z tℓ (˜ γ t ( ξ )) = z tℓ ( γ ( ξ )) − ( ξ − t ) , z ti (˜ γ t ( ξ )) = z ti ( γ ( ξ )) for any i = ℓ. Fix ξ > < t < . . .
Thus, u is admissible for σ I app ( γ, Cε ). Finally, from (42) we get that σ I app ( γ, Cε ) ≤ ε − I ( u, T ) ≤ CT ε − max { k,s } , proving that σ app ( γ, ε ) ε − max { k,s } . This completes the proof. (cid:3) Now, we prove the < part of the statement, in the case where ˙ γ is always contained in the samestratum ∆ k \ ∆ k − . Proposition 7.3.
Assume that there exists s ≥ such that f ⊂ ∆ s \ ∆ s − . Let γ : [0 , T ] → M bea path, such that ˙ γ ( t ) ∈ ∆ k ( γ ( t )) \ ∆ k − ( γ ( t )) for any t ∈ [0 , T ] . Moreover, if s = k , assume that f ( γ ( t )) = ˙ γ ( t ) mod ∆ s − for any t ∈ [0 , T ] . Then, it holds σ I int ( γ, ε ) < σ J int ( γ, ε ) < ε max { s,k } , σ I app ( γ, ε ) < σ J app ( γ, ε ) < ε max { s,k } . Proof.
By Proposition 5.4, σ J int ( γ, ε ) σ I int ( γ, ε ) and σ J app ( γ, ε ) σ I app ( γ, ε ). Hence, to complete theproof it suffices to prove the asymptotic lower bound for σ J int ( γ, ε ) and σ J app ( γ, ε ). In the following,to lighten the notation, we write σ int and σ app instead of σ J int and σ J app . Interpolation by time complexity.
By (38), it suffices to prove that ω ( γ, δ ) < δ { k,s } Let η > u ∈ L ([0 , T ] , R m ) be a control admissible for ω ( γ, δ ) such that(43) J ( u, T ) = k u k L ([0 ,T ] , R m ) ≤ ω ( γ, δ ) δ + η. Let N = ⌈ T /δ ⌉ and 0 = t < t < . . . < t N = T be times such that q u ( t i ) = γ ( t i ), i = 0 , . . . , N ,and 0 < t i − t i − ≤ δ . Observe that, up to removing some t i ’s, we can always assume δ/ ≤ i − t i − ≤ (3 / δ and N ≥ ⌈ (2 T ) / (3 δ ) ⌉ . Moreover, let u i = u | [ t i − ,t i ] . Proceding as in the proof ofTheorem 1.2, p. 22, we get that in order to show that ω ( γ, δ ) < δ { k,s } it suffices to prove(44) k u i k L ([ t i − ,t i ]) ≥ Cδ { s,k } , i = 1 , . . . , N. We distinguish three cases.
Case 1 k > s : Let { z t } be the continuous coordinate family for γ adapted to f given byProposition 3.6. Then, since z tℓ ( γ ( · )) = 0 and z tα ( γ ( ξ )) = ξ − t , by Theorem 4.3 it holds(45) δ ≤ ( t i − t i − ) = | z t i − α ( γ ( t i )) | ≤ C k u i k k L ([ t i − ,t i ]) . This proves (44).
Case 2 k < s : Also in this case, let { z t } be the continuous coordinate family for γ adaptedto f given by Proposition 3.6. Then, by Lemma 4.9 we get δ ≤ t i − t i − ≤ C k u i k s L ([ t i − ,t i ]) , which immediately proves (44). Case 3 k = s : Let { z t } t ∈ [0 ,T ] be a continuous coordinate family for γ adapted to f . By themean value theorem there exists ξ ∈ [ t i − , t i ] such that(46) z t i − ℓ ( γ ( t i )) = Z t i t i − ( z tℓ ) ∗ ˙ γ ( t ) dt = (cid:0) ( z t i − ℓ ) ∗ ˙ γ ( ξ ) (cid:1) ( t i − t i − ) . Consider the partition { E , E , E } of [0 , T ] given by Lemma 3.5 and let δ ≤ δ . Then,depending to which E j belongs t i − , we proceed differently.(a) t i − ∈ E : By Lemma 4.9 and (46) we get t i − t i − ≤ C k u i k s L ([ t i − ,t i ]) + z t i − ℓ ( γ ( t i )) + = C k u i k s L ([ t i − ,t i ]) + (cid:0) ( z t i − ℓ ) ∗ ˙ γ ( ξ ) (cid:1) ( t i − t i − ) . Then, by (21) of Lemma 3.5, we get k u i k L ([ t i − ,t i ]) ≥ − ( z t i − ℓ ) ∗ ˙ γ ( ξ ) C ! s ( t i − t i − ) s ≥ (cid:16) ρC (cid:17) s δ s . This proves (44).(b) t i − ∈ E : By (22) of Lemma 3.5, (46) and Theorem 4.3 we get m ( t i − t i − ) ≤ | z t i − α ( γ ( t i )) | ≤ C (cid:16) k u i k s L ([ t i − ,t i ]) + k u i k L ([ t i − ,t i ]) | z t i − ℓ ( γ ( t i )) | (cid:17) . Reasoning as in (41) yields that we can assume k u i k L ([ t i − ,t i ]) ≤ Cδ s . Then, by (46)and letting δ ≤ ( m/ (2 + 4 ρ )) s , we get k u i k L ([ t i − ,t i ]) ≥ ( m − δ s (1 + 2 ρ )) s ( t i − t i − ) s ≥ (cid:16) m (cid:17) s δ s , proving (44).(c) t i − ∈ E : By Theorem 4.3 it follows that(47) | z t i − ℓ ( γ ( t i )) | ≤ C k u i k s L ([ t i − ,t i ]) + ( t i − t i − ) . Then, by (46) and (47) we obtain k u i k L ([ t i − ,t i ]) ≥ ( z t i − ℓ ) ∗ ˙ γ ( ξ ) − C ! s ( t i − t i − ) s ≥ (cid:16) ρC (cid:17) s δ s . he last inequality follows from (23) of Lemma 3.5. This proves (44). Neighboring approximation complexity.
Fix η > u ∈ L ([0 , T ] , R m ) be admissiblefor σ app ( γ, ε ) and such that k u k L ([0 ,T ] , R m ) ≤ σ app ( γ, ε ) + η . Let q u : [0 , T ] → M be the trajectoryof u with q u (0) = γ (0). Let then N = ⌈ σ app ( γ, ε ) + η ⌉ and 0 = t < t < . . . < t N = T be such that k u k L ([ t i − ,t i ]) ≤ ε for any i = 1 , . . . , N . By Proposition 4.2 and the fact that q u ( t ) ∈ B SR ( γ ( t ) , ε ) forany t ∈ [0 , T ], we can build a new control, still denoted by u , such that q u ( t i ) = γ ( t i ), i = 1 , . . . , N ,and k u k L ([ t i − ,t i ]) ≤ ε .Fixed a δ >
0, w.l.o.g. we can assume that t i − t i − ≤ δ . In fact, we can split each interval[ t i − , t i ] not satisfying this property as t i − = ξ < . . . < ξ M = t i , with ξ ν − ξ ν − ≤ δ . Then, asabove, it is possible to modify the control u so that q u ( ξ ν ) = γ ( ξ ν ) for any ν = 1 , . . . , M . Since M ≤ ⌈ T /δ ⌉ and q u ( · ) ∈ B SR ( γ ( · ) , ε ), we have k u k L ([ ξ i ,ξ i − ]) ≤ ε and the new total number ofintervals is ≤ (1 + ⌈ T /δ ⌉ ) ⌈ σ app ( γ, ε ) + η ⌉ ≤ C ( σ app ( γ, ε ) + η ).We claim that to prove σ app ( γ, ε ) < ε − max { s,k } , it suffices to show that there exists a constant C >
0, independent of u , such that(48) t i − t i − ≤ Cε max { s,k } , for any i = 1 , . . . , N. In fact, since N ≤ C ( σ app ( γ, ε ) + η ), this will imply that T = N X i =1 t i − t i − ≤ C ( σ app ( γ, ε ) + η ) ε max { s,k } . Letting η ↓
0, we get that σ app ( γ, ε ) < ε − max { s,k } , proving the claim.We now let δ sufficiently small in order to apply Lemma 3.5, Theorem 4.3, and Lemma 4.9. Asbefore, we distinguish three cases. Case 1 k > s : Let { z t } be the continuous coordinate family for γ adapted to f given byProposition 3.6. By Theorem 4.3, using the fact that γ ( t i ) = q u ( t i ) for i = 1 , . . . , N , we get( t i − t i − ) = | z t i − α ( γ ( t i )) | ≤ Cε k . (49) This proves (48). Case 2 k < s : Again, let { z t } be the continuous coordinate family for γ adapted to f givenby Proposition 3.6. As for the interpolation by time complexity, by Lemma 4.9 and the factthat q u ( t i ) = γ ( t i ), we get ( t i − t i − ) ≤ Cε s , thus proving (48). Case 3 k = s : Let { z t } t ∈ [0 ,T ] to be a continuous coordinate family for γ adapted to f . Con-sider the partition { E , E , E } of [0 , T ] given by Lemma 3.5 and recall (46). We distinguishthree cases.(a) t i − ∈ E : By Lemma 4.9 and (46) we get t i − t i − ≤ Cε s + z t i − ℓ ( γ ( t i )) = 2 Cε s + (cid:0) ( z t i − ℓ ) ∗ ˙ γ ( ξ ) (cid:1) ( t i − t i − ) . By (21) of Lemma 3.5, this implies t i − t i − ≤ C − ( z t i − ℓ ) ∗ ˙ γ ( ξ ) ! ε s ≤ Cρ ε s . Hence, (48) is proved.(b) t i − ∈ E : By (22) of Lemma 3.5, (46) and Theorem 4.3 we get m ( t i − t i − ) ≤ | z t i − α ( γ ( t i )) | ≤ C (cid:16) ε s + ε | z t i − ℓ ( γ ( t i )) | (cid:17) ≤ C (cid:0) ε s + ε s +1 + ε ( t i − t i − ) (cid:1) . his, by taking ε sufficiently small and enlarging C , implies (48).(c) t i − ∈ E : By Theorem 4.3 it follows that(50) | z t i − ℓ ( γ ( t i )) | ≤ Cε s + ( t i − t i − ) . Then, by (46) and (50) we obtain t i − t i − ≤ C ( z t i − ℓ ) ∗ ˙ γ ( ξ ) − ε s ≤ Cρ ε s . The last inequality follows from (23) of Lemma 3.5, and proves (48). (cid:3)
As for the case of curves, in order to extend Proposition 7.3 to paths not always tangent to thesame strata, we will need the following sub-additivity property. Let us remark that due to thedefinition of the path complexities, we do not need to make any assumption regarding cusps.
Proposition 7.4.
Let γ : [0 , T ] → M be a path and let t , t ⊂ [0 , T ] .i. If there exists k ∈ N such that σ J int ( γ | [ t ,t ] , ε ) < ε − k , then σ J int ( γ, ε ) < ε − k ,ii. σ J app ( γ | [ t ,t ] , ε ) σ J app ( γ, ε ) .Proof. Time interpolation complexity. By (38), it suffices to prove that ω J ( γ | [ t ,t ] , δ ) ω J ( γ, δ ).Let u ∈ L ([0 , T ] , R m ) be a control admissible for Σ J int (Γ , ε ), and let 0 = ξ < . . . < ξ N = T be the times where q u ( ξ i ) = γ ( ξ i ). Let i = i such that t ≤ ξ i ≤ t for any i ∈ { i , . . . , i } .Observe that, by Theorems 2.3 and 4.1, we have V J ( γ ( t ) , γ ( ξ i )) ≤ d SR ( γ ( t ) , γ ( ξ i )) ≤ Cδ r andV J ( γ ( ξ i ) , γ ( t )) ≤ d SR ( γ ( ξ i ) , γ ( t )) ≤ Cδ r , where δ is sufficiently small, C is independent of δ ,and r is the nonholonomic degree of the distribution. Thus, assuming w.l.o.g. C ≥ ω J ( γ | [ t ,t ] , δ ) ≤ δ J ( u | [ t i ,t i ] ) + 2 Cδ r ≤ Cδ J ( u ) + Cδ r . Taking the infimum over all controls u admissible for ω J ( γ, δ ), and recalling that, by Proposition 7.2,it holds ω J ( γ, δ ) δ r , completes the proof. Neighboring approximation complexity.
In this case, the proof is identical to the one of Proposi-tion 6.6 for the tubular approximation complexity. The sole difference is that here, by definition of σ J app , we do not need to assume the absence of cusps. (cid:3) We can now complete the proof of Theorem 1.3, by proving Theorem 7.1.
Proof.
The proof is analogous to the one of Theorem 6.1, using Propositions 7.2, 7.3 and 7.4. (cid:3)
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