Infinite dimensional sub-Riemannian geometry Sylvain Arguill` ere - - PowerPoint PPT Presentation

Finite dimensions An infinite dimensional example Infinite dimensions

Infinite dimensional sub-Riemannian geometry

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, University of Vienna

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Plan

1

Finite dimensions

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An infinite dimensional example

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Infinite dimensions

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Definition

Definition A sub-Riemannian structure on a smooth manifold M is a subbundle ∆ ⊂ TM together with a Riemannian metric g on ∆. A curve t ∈ [0, 1] → q(t) of Sobolev class H1 is horizontal if, for almost every t, ˙ q(t) ∈ ∆q(t), ˙ q(t) ∈ ∆q(t). Its length and action are respectively given by L(q(·)) = 1

• gq(t)( ˙

q(t), ˙ q(t))dt, A(q(·)) = 1 2 1 gq(t)( ˙ q(t), ˙ q(t))dt The sub-Riemannian distance d, and sub-Riemannian geodesics, are defined as usual. A vector field X ∈ Γ(TM) is horizontal if X(q) ∈ ∆q for every q ∈ M. Locally: orthonormal frame of horizontal vector fields X1, . . . , Xk. Horizontal curves: ˙ q(t) = ui(t)Xi(t), ui ∈ L2 and A(q) = 1

2

1

0 ui(t)2dt.

Horizontal vector fields X(q) = ui(q)Xi(q), where u : M → R.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Controllability and the Chow-Rashevski theorem

Let ∆1 = ∆, ∆i+1 = [∆, ∆i] + ∆i, i ≥ 1, and L(∆) =

• i≥1

∆i. Then L ⊂ Γ(TM) be the Lie algebra generated by ∆, i.e., by smooth and horizontal vector fields. Let ki(q) = dim(∆i

q/∆i−1 q

). Theorem (Chow-Rashevski) For M connected and L = TM, any two points of M can be joined by a horizontal curve (controllability). Moreover, for every q0 ∈ M, there are local coordinates q = (x1, . . . , xr) ∈ Rk1 × · · · × Rkr and constants C, C′ > 0 such that C

• |x1|2 + |x2| + · · · +

r

• |xr|2
• ≤ d(q0, q)2 ≤ C′
• |x1|2 + |x2| + · · · +

r

• |xr|2
• In particular, the topology induced by d coincides with its intrisic manifold topology.

Bella¨ ıche 1996 [BR96], Montgomery 2002 [Mon02] Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Endpoint map

Ωq0 : set of all horizontal curves starting at q0 with finite action. Ωq0 smooth Hilbert manifold. Endpoint mapping: E : Ωq0 → M defined by E(q(·)) = q(1): Smooth map. Reachable set from q0: R(q0) = E(Ωq0). Ωq0,q1 = E−1({q1}) may not be a manifold.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

PMP

If q(·) geodesic then (A, E) : Ωq0 → R × M not submersion at q(·): λdA(q(·)) = dE(q(·))∗p1, λ ∈ {0, 1}, p1 ∈ T ∗

q1M.

Hamiltonian characterizations = ⇒ Pontryagin Maximum principle: λ = 1: normal geodesic ⇒ Hamiltonian geodesic equation ( ˙ q, ˙ p) = ∇ωH(q, p)

• n T ∗M, where

H(q, p) = max

u∈∆q

• p(u) − 1

2 gq(u, u)

• = 1

2

k

• i=1

p(Xi(q))2. ⇒ Geodesic flow and cotangent exponential on T ∗M. λ = 0: abnormal geodesic (singular curve) ⇒ abnormal Hamiltonian characterization.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Heisenberg group

M = R3, structure such that X(x, y, z) =

• 1, 0, − y

2

• ,

Y(x, y, z) =

• 0, 1, x

2

• is an orthonormal frame.

Then, for some C, C′ > 0, C(|x|2 + |y|2 + |z|) ≤ d(0, (x, y, z))2 ≤ C′(|x|2 + |y|2 + |z|). Cotangent exponential map: exp0(px, 0, pz) = px pz

• sin(pz), 1 − cos(pz), px − px sin(pz)

pz

• .

Note that exp0(px, py, pz) = (0, 0, z), z = 0 implies pz = 2kπ, k ∈ Z \ {0}.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

In infinite dimensions: Still a distinction between the strong and weak cases for sub-Riemannian Banach manifolds. But even the strong case presents several significant difficulties preventing the generalization of certain finite dimensional sub-Riemannian and/or Riemannian results. In particular, the Pontryagin maximum principle is more complex. It needs to be reformulated as there are geodesics that are neither normal nor abnormal.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Plan

1

Finite dimensions

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An infinite dimensional example

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Infinite dimensions

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Infinite product of Heisenberg groups

Take M = ℓ2(N, R3) = (ℓ2)3, with Hilbert sub-Riemannian structure generated by the

• rthonormal Hilbert frame

Xn(x, y, z) =

• 1, 0, − yn

2

• ,

Yn(x, y, z) =

• 0, 1, xn

2

• ,

n ∈ N. A curve t → (xn(t), yn(t), zn(t))n∈N is horizontal iff each triple (xn(·), yn(·), zn(·)) is horizontal in the 3d-Heisenberg group. Moreover, C(|x|2

ℓ2 + |y|2 ℓ2 + |z|ℓ1) ≤ d(0, (x, y, z))2 ≤ C′(|x|2 ℓ2 + |y|2 ℓ2 + |z|ℓ1).

⇒ R(0) = (ℓ2)2 × ℓ1 ⊂ M: approximate controllability. There are no curves for which dE(q(·)) is surjective: its image is either of positive codimension (singular curves) or dense.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Elusive geodesics

For (zn) ∈ ℓ1 the “obvious” cotangent exponential: for p = (px

n, 0, pz n)n ∈ T ∗ 0 M = (ℓ2)3,

exp0((px

n, 0, pz n)n) =

px

n

pz

n

• sin(pz

n), 1 − cos(pz n), px − px n sin(pz n)

pz

n

• n∈N

, never reaches (0, 0, zn)n∈N ∈ R(0) if each zn = 0: it requires pz

n ≥ 2π for every n.

It is easy to give an explicit geodesic from 0 to (0, 0, zn)n∈N (simply work triplet by triplet). This geodesic is not singular if none of the zn vanishes, and does not appear in the geodesic flow, so it is not normal either. Such geodesics are neither normal or abnormal: they are called elusive. Here, one can obtain them by extending exp0 so that (pz

n) ∈ ℓ∞ (i.e., to

ℓ2 × ℓ∞ = T ∗

0 R(0) (not always true!).

= ⇒ Elusive geodesic comes in part from an incompatibility between the manifold topology and the sub-Riemannian distance. Remark: the curve t ∈ [0, 1] → exp0(tp) is defined even when pz

n → ∞, and is a

critical point of A with fixed endpoints. However, this curve is nowhere minimizing.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Plan

1

Finite dimensions

2

An infinite dimensional example

3

Infinite dimensions

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

D´ efinition

Definition A sub-Riemannian structure on a Banach manifold M is an immersed Banach subbundle ∆ of TM together with a positive definite metric tensor g. Horizontal curves, vector fields, length, action and distance are defined as in finite

• dimensions. Geodesics can be either local or global.

The structure is called strong when g defines a Hilbert norm on each fiber, and weak in all other cases. For strong structures, the sub-riemannian distance topology is at least as fine as the manifold topology. Note that ∆ need not be closed, but it should have a smooth Banach bundle structure such that the inclusion map ∆ ֒ → TM is a smooth bundle morphism.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Chow-Rashevski

Let L ⊂ Γ(TM) be the Lie algebra of vector fields generated by smooth and horizontal vector fields. Theorem (Conditions for approximate controllability) If M is connected and if L is dense in TM then every reachable set is dense.

Dudnikov Samborskii, Grong Markina Vasil’ev

Let X1, . . . , Xr be smooth horizontal vector fields. For I = (i1, . . . , ik) define XI = [Xik , [Xik−1, . . . , [Xi2, Xi1] . . . ]. We say that the structure satisfies the boundedly bracket generating at q ∈ M if, ∃n ∈ N, ∀Z(q) ∈ TqM, Z(q) =

n

• k=0
• I∈{1,...,r}k

[YI, XI](q) with each YI a horizontal vector field. Theorem Assume M is connected and the structure satisfies the strong Chow-Rashevski theorem at every q ∈ M. Then M = R(q), M is locally a finite product of ∆q, and the topology induced by the sub-Riemannian distance is coarser than the intrinsic manifold topology (for strong structure, the topologies coincide, and M is Hilbert).

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Actually, we can find C1-coordinates q = (x1, . . . , xr) ∈ H1 × · · · × Hr, with each Hi a finite product of copies of ∆q0, and C, C′ > 0 such that C(gq(x1, x1) + · · · + gq(xr, xr)1/r) ≤ d(q0, q). Open problem : other side of the ball-box inequality

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Examples

M = R × ℓ2(N, R2), and horizontal vector fields generated by X(x, yn, zn) = ∂ ∂x , Xn(x, yn, zn) = ∂ ∂yn + x ∂ ∂zn . Then, any Z ∈ T0M can be written Z = a ∂ ∂x (0) +

• n∈N

bn ∂ ∂yn (0) + cn ∂ ∂zn (0) = aX + Y1 + [X, Y2], with Y1 = bnXn, Y2 = cnXn. Let M = Ds(N), (N, ∆0, g0) a compact connected sub-Riemannian manifold with ∆0 bracket-generating, s > dim(N)/2 + 1. The sub-Riemannian structure

• n Ds(N) such that horizontal curves are the flows of horizontal vector fields

satisfy the strong Chow-Rashevski property (non-trivial).

Agrachev Caponigro, Arguill` ere Tr´ elat. Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Endpoint map

Ωq0 : set of all horizontal curves starting at q0 of class H1. Ωq0 may not be globally a smooth manifold (requires a local addition on M). However, has a smooth structure in the neighbourhood U of a constant curves. Endpoint mapping: E : Ωq0 → M defined by E(q(·)) = q(1): Smooth map on U. Ωq0,q1 ∩ U = E−1({q1}) ∩ U may still not be a manifold. Lemma (PMP) Let q(·) ∈ Ωq0,q1 ∩ U be a minimizing geodesic. Then (dA(q(·), dE(q(·)) is not surjective, so that one of the following statements is true:

1

∃(λ, p1) ∈ {0, 1} × T ∗

q1M \ {(0, 0)}, λdA(q(·)) = dE(q(·))∗p1. If λ = 1: normal

geodesic. If λ = 0, q(·) is a singular point of E: abnormal geodesic. ⇒ Hamiltonian characterization.

2

Im(dA(q(·)), dE(q(·)) is a proper dense subset of R × Tq1M (elusive geodesic) ⇒ No Hamiltonian characterization.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Partial converse and Hamiltonian

Partial converse: Lemma Let q(·) ∈ Ωq0,q1 ∩ U and p1 ∈ T ∗

q1M. 1

If dA(q(·)) = dE(q(·))∗p1 then q(·) is a critical point of the action with fixed endpoints.

2

If p1 = 0 and 0 = dE(q(·))∗p1, then q(·) is a critical point of E. We say that q is an abnormal curve. Obviously, not every critical point of A with fixed endpoints have this form. However, these condition can be interpreted using a Hamiltonian. Define Hλ : T ∗M ⊕ ∆ → R : Hλ(q, p, u) = (p | u) − λ 2 gq(u, u).

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Hamiltonian formulation

Let ω be the canonical weak symplectic form of T ∗M, and λ ∈ {0, 1}. If ∂uHλ(q, p, u) = 0, then ∇ωHλ(q, p, u) = (∂pH, −∂qH) ∈ T(q,p)T ∗M is defined intrinsically. Proposition λdA(q(·)) = dE(q(·))∗p1 ⇐ ⇒ ∃t → p(t) ∈ T ∗

q(t)M, p(1) = p1,

• = ∂uHλ(q(t), p(t), u(t)),

( ˙ q(t), ˙ p(t)) = ∇ωHλ(q(t), p(t), u(t)) = (∂pHλ, −∂qHλ). For λ = 0, (q, u) singular point of E: abnormal curve. For λ = 1, q extremal point of the action: p(·) is the momentum of q. ∂uH1(q, p, u) = 0 ⇐ ⇒ gq(u, ·) = p|∆ ∈ ∆∗

q

⇔ H1(q, p, u) = max

u′∈Hq

H1(q, p, u′) := H(q, p). The mapping H : T ∗M → R ∪ {+∞} is the normal Hamiltonian.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Geodesic flow: strong structures

Let Kq : ∆∗

q → ∆q such that ∀α ∈ ∆∗ q, α = gq(Kqα, ·). K smooth vector bundle

• morphism. Then

gq(u, ·) = p|∆q ⇐ ⇒ u = Kqp|∆q , so that H(q, p) = 1

2 pKqp|∆q is as smooth as ξ. Then ∂pH(q, p) ∈ T ∗∗ q M can be

identified to Kqp|∆q ∈ TqM. = ⇒ H possesses a smooth symplectic gradient on T ∗M. Theorem (Hamiltonian geodesic flow) For a strong structure, for any initial condition (q(0), p(0)) ∈ T ∗M, there exists a unique maximal solution t → (q(t), p(t)) ∈ T ∗M to the equation ( ˙ q(t), ˙ p(t)) = ∇ωH(q(t), p(t)) = (∂pH(q, p), −∂qH(q, p)). Then q(·) is a (normal) global geodesic.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Elusive geodesics

Remark: By restricting the structure to a stable embedded dense submanifold M′ ⊂ M, we obtain bigger cotangent spaces and hence ”more” normal (and abnormal)

• geodesics. Those geodesics were elusive in M: they didn’t appear in the geodesic flow.

Open problem: is there an “ideal” cotangent space (such that elusive geodesics disappear)? Example: on the infinite product of Heisenberg groups, as O0 = ℓ2(N, R2) × ℓ1(N, R), taking p(0) in ℓ2(N, R2) × ℓ∞(N, R) makes all minimizing geodesics into normal geodesics? Conjecture: On the ideal cotangent space, singular curves are exactly the semi-rigid curves of Grong Markina Vasil’ev ⇒ way to find every singular curve without pinpointing that cotangent space.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

The weak case and adapted cotangent sub-bundles

Difficulty: for weak structures, h(q, p) may be infinite. Definition A dense sub-bundle τ ∗M ֒ → T ∗M is said to be adapted to the structure if the restriction of H to τ ∗M is finite. The restriction of ω to such a sub-bundle remains a weak symplectic form. However, the restriction of H may not have a symplectic gradient. Examples: For the Riemannian case H = TM, simply take τ ∗M = g(TM, ·). τ ∗M = g(TM, ·) also works if g can be extended to TM .

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Geodesic flow: weak structures

Theorem Let τ ∗M be an adapted cotangent sub-bundle on which H possesses a C2 symplectic

• gradient. Then for any initial condition (q(0), p(0)) ∈ τ ∗M, there exists a unique

solution t → (q(t), p(t)) ∈ τ ∗M such that ( ˙ q(t), ˙ p(t)) = ∇ωH(q(t), p(t)) = (∂pH(q, p), −∂qH(q, p)). In this case, q(·) is a local geodesic. remarks: For a riemannian structure with τ ∗M = g(TM·), H has a smooth symplectic gradient iff the metric admits a smooth Levi-Civita connection. If g can be extended to TM and τ ∗M = g(TM, ·), then H has a smooth symplectic gradient if and only if the geodesic equations obtained in Grong Markina Vasil’ev have a well-defined flow.

Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry

Finite dimensions An infinite dimensional example Infinite dimensions

Perspectives and other open questions Study of simple examples (Carnot groups) How to rule out elusive geodesics? Applications to groups of diffeomorphisms

Series of works in collaboration:

• S. Arguill`

ere, Infinite dimensional sub-Riemannian geometry, In preparation.

• S. Arguill`

ere, E. Tr´ elat, A. Trouv´ e, L. Younes, Multiple shape registration using constrained optimal control, Preprint (2015).

• S. Arguill`

ere, E. Tr´ elat, Sub-Riemannian structures on groups of diffeomorphisms, Preprint (2014).

• S. Arguill`

ere, E. Tr´ elat, A. Trouv´ e, L. Younes, Shape deformation analysis from the optimal control viewpoint, to appear in J. Math. Pures Appl. (2015). Sylvain Arguill` ere (CIS, Johns Hopkins University) Workshop on infinite dimensional Riemannian geometry, Infinite dimensional sub-Riemannian geometry