On the stabilization problem for nonholonomic distributions
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On the stabilization problem for nonholonomic distributions L. Rifford? E. Trelat† Abstract Let M be a smooth connected and complete manifold of dimension n, and ∆ be a smooth nonholonomic distribution of rank m ≤ n on M . We prove that, if there exists a smooth Riemannian metric on ∆ for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of ∆ onM . Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the cor- responding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories. 1 Introduction Throughout this paper, M denotes a smooth connected manifold of dimension n. 1.1 Stabilization of nonholonomic distributions Let ∆ be a smooth distribution of rankm ≤ n onM , that is, a rankm subbundle of the tangent bundle TM of M . This means that, for every x ? M , there exist a neighborhood Vx of x in M , and a m-tuple (fx1 , . . . , f x m) of smooth vector fields on Vx, linearly independent on Vx, such that ∆(y) = Span {fx1 (y), .
Voir
- unique smooth function
- smooth metric
- distribution ∆ associated
- let x¯ ?
- abstract let
- point mapping
- conjugate locus
- distribution ∆
- minimizing beyond
- singular path
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On the stabilization problem for nonholonomic distributions L. Rifford∗elat.Tr´E†
Abstract LetMbe a smooth connected and complete manifold of dimensionn, and Δ be a smooth nonholonomic distribution of rankm≤nonM. We prove that, if there exists a smooth Riemannian metric on Δ for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of Δ onM. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the cor-responding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories.
Introduction
Throughout this paper,Mdenotes a smooth connected manifold of dimensionn.
1.1 Stabilization of nonholonomic distributions
Let Δ be a smooth distribution of rankm≤nonM, that is, a rankmsubbundle of the tangent bundleT MofM means that, for every. Thisx∈M, there exist a neighborhoodVxofxin M, and am-tuple (f1x, . . . , fmx) of smooth vector fields onVx, linearly independent onVx, such that Δ(y) = Span{f1x(y), . . . , fmx(y)},∀y∈ Vx. One says that them-tuple of vector fields (f1x, . . . , fmx) represents locally the distribution Δ. The distribution Δ is said to benonholonomic(also called totally nonholonomice.g.in [3]) if, for everyx∈M, there is am-tuple (f1x, . . . , fmx) of smooth vector fields onVxwhich represents locally the distribution and such that Lie{fx. . , fxm}(y) =TyM,∀y∈ Vx, 1, . that is, such that the Lie algebra spanned byf1x, . . . , fxm, is equal to the whole tangent space TyM, at every pointy∈ Vx Lie algebra property is often called. Thistionondi¨ro’rsmcandHe. Anhorizontal pathngnioijx0tox1is an absolutely continuous curveγ(∙) : [0,1]→M such thatγ(0) =x0,γ(1) =x1, and such thatγ˙ (t)∈Δ(γ(t)), for almost everyt∈[0,1]. According to the classical Chow-Rashevsky Theorem (see [9, 19, 33, 36]), since the distribution is nonholonomic onM, any two points ofMcan be joined by an horizontal path. Let Δ be a nonholonomic distribution andx¯∈M recall that, for a smooth Webe fixed. vector fieldXonM ˙, the dynamical systemx=X(x) is said to beglobally asymptotically stable at the point ¯x, if the two following properties are satisfied: ∗itopilU,nsiavLeorbsJi.tA´.edeNice-SophiaAn,126craPrlaV,esoie.Donude,n´R6UM,8Nic0610ex02eCed France (ro@dirffni.uthmar.fce) †R6628,RoMAPMO,UM.hL,ba.oaesnM,taOr67eal´596750,4ertrPB,sdetuahCeedxsnectisrevin´lrO’de´U 2, France (lat@.TrenuelEmmarf.snaelro-vinu)
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Lyapunov stability:for every neighborhoodVofx¯, there exists a neighborhoodWofx¯ such that, for everyx∈ W ˙, the solution ofx(t) =X(x(t)), x(0) =x, satisfiesx(t)∈ V, for everyt≥0.
Attractivity:for everyx∈M ˙, the solution ofx(t) =X(x(t)), x(0) =x,tends tox¯ ast tends to +∞.
The stabilization problem for nonholonomic distributions consists in finding, if possible, a smooth stabilizing sectionXof Δ, that is, a smooth vector fieldXonMsisatinfygX(x)∈Δ(x) for everyx∈M, such that the dynamical systemx˙ =X(x) is globally asymptotically stable atx¯. There exist two main obstructions for a distribution to admit a stabilizing section. The first one is of global nature: it is well-known that, if the manifoldMadmits such a dynamical system, then it possesses a smooth Lyapunov function,i.e., a Morse function having only one (possibly degenerate) critical point inM consequence,. InMmust be homeomorphic to the Euclidean space IRn The second one is of local(we refer the reader to [39] for further details). nature: due toBrockett’s condition1, (iii)]; see also [23, 44]), the distribution(see [13, Theorem Δ cannot admit a smooth stabilizing section wheneverm < n. The absence of smooth stabilizing sections motivates to define a new kind of stabilizing section. The first author has recently introduced the notion of smooth repulsive stabilizing feedback for control systems1(see [39, 40, 41]), whose definition can be easily translated in terms of stabilizing section. Let ¯x∈M Letbe fixed.Sbe a closed subset ofMandXbe a vector field onM. The dynamical systemx˙ =X(x) is said to besmooth repulsive globally asymptotically stable atx¯ with respect toS(denoted in short SRS¯x,S) if the following properties are satisfied: (i) The vector fieldXis locally bounded onMand smooth onM\ S.
(ii) The dynamical system ˙x=X(x) is globally asymptotically stable atx¯ in the sense of Carath´eodory,namely,foreveryx∈M, there exists a solution of
˙x(t) =X(x(t)),for almost everyt∈[0,∞), x(0) =x,
(1)
and, for everyx∈Mverysolu,e()aclldeitnofo1(orodolysraCa´ethoitufonx˙ =X(x)) on [0,∞) tends tox¯ asttends to∞ for every neighborhood. Moreover,Vofx¯, there exists a neighborhoodWofx¯ such that, forx∈ W, the solutions of (1) satisfyx(t)∈ V, for everyt≥0.
(iii) For everyx∈M, the solutions of (1) satisfyx(t)/∈ S,for everyt >0.
In view of what happens whenever Δ =T M, and having in mind the above obstructions for the stabilization problem, a natural question is to wonder if, given a smooth nonholonomic distribution Δ, there exists a sectionXof Δ onMand a closed nonempty subsetSofMsuch thatXis SRSx¯,S this paper, we provide a positive answer in a large number of situations.. In To state our main results, we need to endow the distribution Δ with a Riemannian metric, thus encountering the framework of sub-Riemannian geometry, and we require the concept of a singular path, recalled next. 1If one represents locally the distribution Δ by am-tuple of smooth vector fields (f1, , fm), then the ∙ ∙ ∙ existence of a local stabilizing section for Δ is equivalent to the existence of a stabilizing feedback for the associated control system ˙x=Pmi=1uifi(x). There is a large literature on alternative types of stabilizing feedbacks for control systems (see Section 1.4).
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