LOCAL GEOMETRY OF NONREGULAR CARNOT-CARATH EODORY SPACES AND - PowerPoint PPT Presentation

LOCAL GEOMETRY OF NONREGULAR CARNOT-CARATH´ EODORY SPACES AND APPLICATIONS TO NONLINEAR CONTROL THEORY Svetlana Selivanova Sobolev Institute of Mathematics, Novosibirsk Bia� lowieˆ za, Poland, XXXI Workshop on Geometric Methods in Physics 24 June – 30 June, 2012

References Selivanova S.V. Metric geometry of nonregular weighted Carnot- Carath´ eodory spaces, arXiv:1206.6608v1. Selivanova S.V. Local geometry of nonregular weighted quasi- metric Carnot-Caratheodory spaces // Doklady Mathematics, 2012. Vol. 443, No. 1, P. 16–21.

Motivation Geometric control theory . ⋄ The linear system of ODE ( x ∈ M N , m < N ) m � x = ˙ a i ( t ) X i ( x ) (1) i =1 is locally controllable iff Lie { X 1 , X 2 , . . . , X m } = T M , i.e. the “horizontal” distribution H M = { X 1 , X 2 , . . . , X m } is bracket-generating: • span { X I ( v ) : | I | ≤ M } = T v M for all v ∈ M , where X I = [ X i 1 , [ X i 2 , . . . , [ X i k − 1 , X i k ], | I | = k (H¨ ormander’s condition) • M is the depth of the sub-Riemannian space M • Rashevsky-Chow theorem ⇒ on M there exists an intrinsic metric d c ( u, v ) = inf { L ( γ ) } γ − horizontal γ (0)= u,γ (1)= v

Motivation Geometric control theory . ⋄ The linear system of ODE ( x ∈ M N , m < N ) m � x = ˙ a i ( t ) X i ( x ) (2) i =1 is locally controllable iff Lie { X 1 , X 2 , . . . , X m } = T M , i.e. the “horizontal” distribution H M = { X 1 , X 2 , . . . , X m } is bracket-generating: • span { X I ( v ) : | I | ≤ M } = T v M for all v ∈ M , where X I = [ X i 1 , [ X i 2 , . . . , [ X i k − 1 , X i k ], | I | = k (H¨ ormander’s condition) • M is the depth of the sub-Riemannian space M • Rashevsky-Chow theorem ⇒ on M there exists an intrinsic metric d c ( u, v ) = γ − horizontal γ (0)= u,γ (1)= v { L ( γ ) } inf

Motivation Geometric control theory . ⋄ The linear system of ODE ( x ∈ M N , m < N ) m � x = ˙ a i ( t ) X i ( x ) (3) i =1 is locally controllable iff Lie { X 1 , X 2 , . . . , X m } = T M , i.e. the “horizontal” distribution H M = { X 1 , X 2 , . . . , X m } is bracket-generating: • span { X I ( v ) : | I | ≤ M } = T v M for all v ∈ M , where X I = [ X i 1 , [ X i 2 , . . . , [ X i k − 1 , X i k ], | I | = k (H¨ ormander’s condition) • M is the depth of the sub-Riemannian space M • Rashevsky-Chow theorem ⇒ on M there exists an intrinsic metric d c ( u, v ) = inf { L ( γ ) } γ − horizontal γ (0)= u,γ (1)= v

• Filtration H M = H 1 ⊆ H 2 ⊆ . . . ⊆ H M = T M such that [ H 1 , H i ]= H i +1 (H¨ ormander’s condition). Here H k ( v ) = span { [ X i 1 , [ X i 2 , . . . , [ X i k − 1 , X i k ] . . . ]( v ) : X i j ∈ H 1 } • A point u ∈ M is called regular if dim H k ( v ) = const in some neighborhood v ∈ U ( u ) ⊆ M . Otherwise, u is called nonregular.

• Examples. Regular: Heisenberg groups, Carnot groups, roto- translation group, etc. Nonregular: Groushin-type planes (related to the PDE ∂ 2 u ∂x 2 + x 2 k∂ 2 u ∂x 2 = f ) M = R 2 . H 1 = span { X 1 = ∂ ∂x , X 2 = x k ∂ ∂y } . The axis x = 0 consists of nonregular points; the depth is M = k + 1. There are no regular C-C structures on R 2 !

⋄ (J.-M. Coron, etc.) The sufficient condition of controllability of the nonlinear system   . x = f ( x, a ) , ˙ (4)  x (0) = x 0 , is that � h (0) : h ∈ Lie ∂ | α | ∂a α f (0 , · ) , α ∈ N M � span = T x 0 M for some M ∈ N . Letting � ∂ α � F ν = ∂a α f (0 , · ) : | α | ≤ ν and H k ( q ) = span { [ X 1 , [ X 2 , . . . , [ X i − 1 , X i ] . . . ]( q ) : X j ∈ F ν j , ν 1 + ν 2 + . . . + ν i ≤ k } , one obtains a weighted filtration H 1 ⊆ H 2 ⊆ . . . ⊆ H M = T M , such that [ H i , H j ] ⊆ H i + j more general than the H¨ ormander condition

⋄ (J.-M. Coron, etc.) The sufficient condition of controllability of the nonlinear system   . x = f ( x, a ) , ˙ (5)  x (0) = x 0 , is that � h (0) : h ∈ Lie ∂ | α | ∂a α f (0 , · ) , α ∈ N M � span = T x 0 M for some M ∈ N . Letting � ∂ α � F ν = ∂a α f (0 , · ) : | α | ≤ ν and H k ( q ) = span { [ X 1 , [ X 2 , . . . , [ X i − 1 , X i ] . . . ]( q ) : X j ∈ F ν j , ν 1 + ν 2 + . . . + ν i ≤ k } , one obtains a weighted filtration H 1 ⊆ H 2 ⊆ . . . ⊆ H M = T M , such that [ H i , H j ] ⊆ H i + j more general than the H¨ ormander condition

Some references concerning the underlying geometry • Nagel, Stein, Wainger 1985; • Gromov 1996; • Coron 1996; • Christ, Nagel, Stein, Wainger 1999; • Rampazzo, Sussmann 2001, 2007 • Tao, Wright 2003 • Agrachev, Marigo 2003; • Montanari, Morbidelli 2004, 2011; • Street 2011 • Karmanova, Vodopyanov 2007–2009; Karmanova 2010, 2011.

Weighted Carnot-Carath´ eodory spaces • M , dim M = N is a smooth connected manifold . • X 1 , X 2 , . . . , X q ∈ C 2 M +1 span T M ; deg X i := d i , d 1 ≤ . . . ≤ d q . • X I = [ X i 1 , [ . . . , [ X i k − 1 , X i k ] . . . ], where I = ( i 1 , . . . , i k ); | I | h := d i 1 + . . . + d i k . • H j = span { X I | | I | h ≤ j } . H M = H 1 ⊆ H 2 ⊆ . . . ⊆ H M = T M [ H i , H j ] ⊆ H i + j . Here [ H i , H j ] is the linear span of commutators of the vector field generating H i and H j . Model case: d 1 := 1 , d q := M .

Problems 1. In a neighborhood of a nonregular point, the basis Y 1 , Y 2 , . . . , Y N , . associated to the filtration H 1 ⊆ H 2 ⊆ . . . ⊆ H M , varies discon- tinuously from point to point. 2. In the case of a weighted filtration the intrinsic Carnot- Carath´ eodory metric d c might not exist. Example (Stein “Harmonic Analysis”) M = R N with standard basis ∂ x 1 , ∂ x 2 , . . . , ∂ x N . Let deg( ∂ x i ) = 1 for 1 ≤ i ≤ m ; deg( ∂ x i ) > 1 for i > m . Evidently, H i = span { ∂ x 1 , ∂ x 2 , . . . , ∂ x i } satisfy [ H i , H j ] ⊆ H i + j , since [ H i , H j ] = { 0 } . But H 1 = span { ∂ x i } m i =1 (for any m < N ) does not span R N .

3. Different choices of weights may lead to different combina- tions of regular and nonregular points. . Example M = R 3 ; vector fields { X 1 = ∂ y , X 2 = ∂ x + y∂ t , X 3 = ∂ x } . Nontrivial commutator: [ X 1 , X 2 ] = ∂ t . 1. Let deg( X i ) := 1, i = 1 , 2 , 3. Then deg([ X 1 , X 2 ]) = 2 and H 1 = span { X 1 , X 2 , X 3 } , H 2 = H 1 ∪ span { [ X 1 , X 2 ] } . In this case { y = 0 } is a plane consisting of nonregular points. 2. Let deg( X 1 ) := a, deg( X 2 ) := b, deg( X 3 ) := a + b , a ≤ b . Then deg([ X 1 , X 2 ]) = a + b ⇒ H a = span { X 1 } , H b = H a ∪ span { X 2 } , H a + b = H a ∪ H b ∪ span { X 3 , [ X 1 , X 2 ] } In this case all points of R 3 are regular.

3. Different choices of weights may lead to different combina- tions of regular and nonregular points. . Example M = R 3 ; vector fields { X 1 = ∂ y , X 2 = ∂ x + y∂ t , X 3 = ∂ x } . Nontrivial commutator: [ X 1 , X 2 ] = ∂ t . 1. Let deg( X i ) := 1, i = 1 , 2 , 3. Then deg([ X 1 , X 2 ]) = 2 and H 1 = span { X 1 , X 2 , X 3 } , H 2 = H 1 ∪ span { [ X 1 , X 2 ] } . In this case { y = 0 } is a plane consisting of nonregular points. 2. Let deg( X 1 ) := a, deg( X 2 ) := b, deg( X 3 ) := a + b , a ≤ b . Then deg([ X 1 , X 2 ]) = a + b ⇒ H a = span { X 1 } , H b = H a ∪ span { X 2 } , H a + b = H a ∪ H b ∪ span { X 3 , [ X 1 , X 2 ] } In this case all points of R 3 are regular.

Questions Are there some analogs of classical results of sub-Riemannian geometry for weighted C-C spaces? ⋄ Results on existence and the algebraic structure of the Gro- mov’s tangent cone to M = ( M , d c ) at a fixed point u ∈ M : it is a homogeneous space of a Carnot group ( G/H, d u c ). ⋄ Local approximation theorem: if d c ( u, v ) = O ( ε ) and d c ( u, v ) = c ( v, w ) | = O ( ε 1+ 1 O ( ε ), then | d c ( u, v ) − d u M ). ⋄ Methods of optimal motion planning for the system (1).

Metric structure We work with the following quasimetric Nagel, Stein, Wainger 1985: ρ ( v, w ) = inf { δ > 0 | there is a curve γ : [0 , 1] → U , � w I X I ( γ ( t )) , | w I | < δ | I | h } . γ (0) = v, γ (1) = w, ˙ γ ( t ) = | I | h ≤ M Here X I = [ X i 1 , [ . . . , [ X i k − 1 , X i k ] . . . ], where I = ( i 1 , . . . , i k ); | I | h = d i 1 + . . . + d i k . 1 deg Yi } For the regular case ρ ( v, w ) = d ∞ ( v, w ) = i =1 ,...,N {| v i | max

Quasimetric space ( X, d X ) X is a topoogical space; d X : X × X → R + is such that (1) d X ( u, v ) ≥ 0; d X ( u, v ) = 0 ⇔ u = v ; (2) d X ( u, v ) ≤ c X d X ( v, u ), where 1 ≤ c X < ∞ uniformly on u, v ∈ X (generalized symmetry property); (3) d X ( u, v ) ≤ Q X ( d X ( u, w ) + d X ( w, v )), where 1 ≤ Q X < ∞ uniformly on all u, v, w ∈ X (generalized triangle inequality); (4) d X ( u, v ) upper semicontinuous on the first argument Q X = c X = 1 ⇒ ( X, d X ) metric space

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Basic considerations . • Choice of basis { Y 1 , Y 2 , . . . , Y N } among { X I } | I | h ≤ M : ∗ Y 1 , Y 2 , . . . , Y N are linearly independent at u (hence in some neighborhood U ( u )); N � ∗ deg Y i is minimal; i =1 N � ∗ | I j | is minimal, where Y j = X I j . j =1 • Coordinates of the second kind Φ u : R N → U Φ u ( x 1 , . . . , x N ) = exp( x 1 Y 1 ) ◦ exp( x 2 Y 2 ) ◦ . . . ◦ exp( x N Y N )( u )