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) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions Plan 1 Finite dimensions An infinite dimensional example 2 3 Infinite dimensions Sylvain Arguill` ere (CIS, Johns Hopkins University) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional 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 H 1 is horizontal if, for almost every t, q ( t ) ∈ ∆ q ( t ) , ˙ ˙ q ( t ) ∈ ∆ q ( t ) . Its length and action are respectively given by � 1 � 1 A ( q ( · )) = 1 � g q ( t ) ( ˙ q ( t ) , ˙ g q ( t ) ( ˙ q ( t ) , ˙ L ( q ( · )) = q ( t )) dt , q ( t )) dt 2 0 0 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 X 1 , . . . , X k . q ( t ) = � u i ( t ) X i ( t ) , u i ∈ L 2 and A ( q ) = 1 � � 1 Horizontal curves: ˙ 0 u i ( t ) 2 dt . 2 Horizontal vector fields X ( q ) = � u i ( q ) X i ( q ) , where u : M → R . Sylvain Arguill` ere (CIS, Johns Hopkins University) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions Controllability and the Chow-Rashevski theorem Let ∆ 1 = ∆ , ∆ i + 1 = [∆ , ∆ i ] + ∆ i , � ∆ i . i ≥ 1 , and L (∆) = i ≥ 1 Then L ⊂ Γ( TM ) be the Lie algebra generated by ∆ , i.e., by smooth and horizontal q / ∆ i − 1 vector fields. Let k i ( q ) = dim (∆ i ) . 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 q 0 ∈ M, there are local coordinates q = ( x 1 , . . . , x r ) ∈ R k 1 × · · · × R k r and constants C , C ′ > 0 such that � � � � � � | x 1 | 2 + | x 2 | + · · · + ≤ d ( q 0 , q ) 2 ≤ C ′ | x 1 | 2 + | x 2 | + · · · + r r | x r | 2 | x r | 2 C 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) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions Endpoint map Ω q 0 : set of all horizontal curves starting at q 0 with finite action. Ω q 0 smooth Hilbert manifold. Endpoint mapping: E : Ω q 0 → M defined by E ( q ( · )) = q ( 1 ) : Smooth map. Reachable set from q 0 : R ( q 0 ) = E (Ω q 0 ) . Ω q 0 , q 1 = E − 1 ( { q 1 } ) may not be a manifold. Sylvain Arguill` ere (CIS, Johns Hopkins University) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions PMP If q ( · ) geodesic then ( A , E ) : Ω q 0 → R × M not submersion at q ( · ) : λ dA ( q ( · )) = dE ( q ( · )) ∗ p 1 , λ ∈ { 0 , 1 } , p 1 ∈ T ∗ q 1 M . Hamiltonian characterizations = ⇒ Pontryagin Maximum principle: λ = 1: normal geodesic ⇒ Hamiltonian geodesic equation ( ˙ q , ˙ p ) = ∇ ω H ( q , p ) on T ∗ M , where k � p ( u ) − 1 � = 1 � p ( X i ( q )) 2 . H ( q , p ) = max 2 g q ( u , u ) 2 u ∈ ∆ q i = 1 ⇒ Geodesic flow and cotangent exponential on T ∗ M . λ = 0: abnormal geodesic (singular curve) ⇒ abnormal Hamiltonian characterization. Sylvain Arguill` ere (CIS, Johns Hopkins University) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions Heisenberg group M = R 3 , structure such that 1 , 0 , − y 0 , 1 , x � � � � X ( x , y , z ) = , Y ( x , y , z ) = 2 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: sin ( p z ) , 1 − cos ( p z ) , p x − p x sin ( p z ) exp 0 ( p x , 0 , p z ) = p x � � . p z p z Note that exp 0 ( p x , p y , p z ) = ( 0 , 0 , z ) , z � = 0 implies p z = 2 k π, k ∈ Z \ { 0 } . Sylvain Arguill` ere (CIS, Johns Hopkins University) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional 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) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions Plan 1 Finite dimensions An infinite dimensional example 2 3 Infinite dimensions Sylvain Arguill` ere (CIS, Johns Hopkins University) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions Infinite product of Heisenberg groups Take M = ℓ 2 ( N , R 3 ) = ( ℓ 2 ) 3 , with Hilbert sub-Riemannian structure generated by the orthonormal Hilbert frame 1 , 0 , − y n 0 , 1 , x n � � � � X n ( x , y , z ) = , Y n ( x , y , z ) = , n ∈ N . 2 2 A curve t �→ ( x n ( t ) , y n ( t ) , z n ( t )) n ∈ N is horizontal iff each triple ( x n ( · ) , y n ( · ) , z n ( · )) is horizontal in the 3d-Heisenberg group. Moreover, ℓ 2 + | z | ℓ 1 ) ≤ d ( 0 , ( x , y , z )) 2 ≤ C ′ ( | x | 2 C ( | x | 2 ℓ 2 + | y | 2 ℓ 2 + | y | 2 ℓ 2 + | z | ℓ 1 ) . ⇒ R ( 0 ) = ( ℓ 2 ) 2 × ℓ 1 ⊂ 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) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions Elusive geodesics For ( z n ) ∈ ℓ 1 the “obvious” cotangent exponential: for p = ( p x n , 0 , p z n ) n ∈ T ∗ 0 M = ( ℓ 2 ) 3 , � p x n ) , p x − p x n sin ( p z � n ) �� exp 0 (( p x n , 0 , p z n sin ( p z n ) , 1 − cos ( p z n ) n ) = , p z p z n n n ∈ N never reaches ( 0 , 0 , z n ) n ∈ N ∈ R ( 0 ) if each z n � = 0: it requires p z n ≥ 2 π for every n . It is easy to give an explicit geodesic from 0 to ( 0 , 0 , z n ) n ∈ N (simply work triplet by triplet). This geodesic is not singular if none of the z n 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. n ) ∈ ℓ ∞ (i.e., to Here, one can obtain them by extending exp 0 so that ( p z ℓ 2 × 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 ] �→ exp 0 ( tp ) is defined even when p z n → ∞ , and is a critical point of A with fixed endpoints. However, this curve is nowhere minimizing. Sylvain Arguill` ere (CIS, Johns Hopkins University) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,

Finite dimensions An infinite dimensional example Infinite dimensions Plan 1 Finite dimensions An infinite dimensional example 2 3 Infinite dimensions Sylvain Arguill` ere (CIS, Johns Hopkins University) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional 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) Infinite dimensional sub-Riemannian geometry Workshop on infinite dimensional Riemannian geometry,