Subriemannian minimizers H. J. Sussmann Department of Mathematics - PowerPoint PPT Presentation

Subriemannian minimizers H. J. Sussmann Department of Mathematics — Rutgers University Piscataway, NJ 08854, USA sussmann@math.rutgers.edu Conference on Nonlinear Partial Differential Equations and Applications In honor of Jean-Michel Coron’s 60th birthday Paris, June 20-24, 2016

HAPPY BIRT HDAY J EAN − MICHEL !! 2

VECTOR DISTRIBUTIONS AND SECTIONS A vector distribution on a smooth manifold M is a vector subbundle E of the tangent bundle TM of M . (That is, E assigns to each x ∈ M a linear subspace E ( x ) of the tangent space T x M , in such a way that the dimension of E ( x ) is the same for all x ∈ M . The fiber dimension of E is the dimension of the spaces E ( x ). A section of a vector distribution E on M over an open subset Ω of M is a vector field V on Ω such that V ( x ) ∈ E ( x ) for each x ∈ Ω. If κ = ∞ or κ = ω , and M is of class C κ , we use Γ κ ( E, Ω) to denote the set of all sections of E over Ω that are of class C κ . ——————— In this lecture, a. “manifold” means “finite-dimensional paracompact manifold without boundary”, b.“smooth” means “of class C ∞ ”, c. If M is a smooth manifold, then TM , T ∗ M are, respectively, the tangent and cotangent bundles of M . If x ∈ M , then T x M , T ∗ x M are, respectively, the tangent and cotangent spaces of M at x . 3

SMOOTH AND REAL ANALYTIC DISTRIBUTIONS The vector distribution E is smooth if E is a smooth submanifold (Equivalently, E is smooth iff for every x ∈ M and every of TM . v ∈ E ( x ) there exists a smooth section V on some neighborhood Ω of x such that V ( x ) = v .) If M is real analytic, then the vector distribution E is real analytic if E is a real analytic submanifold of TM . (Equivalently, E is real analytic iff for every x ∈ M and every v ∈ E ( x ) there exists a real analytic section V on some neighborhood Ω of x such that V ( x ) = v .) A TRIVIAL WELL-KNOWN FACT: If E is a smooth (or real ana- lytic) distribution on M with fiber dimension d , then for every x ∈ M there exists a local basis of smooth (or real analytic) sections of E near x , that is, a d -tuple ( X 1 , . . . , X d ) of smooth (or real analytic) sections of E defined on an open neighborhood Ω of x such that ( X 1 ( y ) , . . . , X d ( y )) is a basis of E ( y ) for every y ∈ Ω. 4

GLOBAL SECTIONS A global section of E is a section of E over M . A WELL-KNOWN FACT: If E is a smooth distribution on M and dim M = n , then E has 2 n +1 smooth global sections X 1 , . . . , X 2 n +1 such that the vectors X 1 ( x ) , . . . , X 2 n +1 ( x ) linearly span E ( x ) for each x ∈ M . (Proof: Use Whitney’s mbedding theorem.) ANOTHER WELL-KNOWN FACT: If E is a real analytic distribu- tion on the real analytic manifold M and dim M = n , then E has 4 n + 2 real analytic global sections X 1 , . . . , X 4 n +2 such that the vec- tors X 1 ( x ) , . . . , X 4 n +2 ( x ) linearly span E ( x ) for each x ∈ M . (Proof: Use the Morrey-Grauert embedding theorem.) 5

RIEMANNIAN METRICS A smooth Riemannian metric on a smooth distribution E on a smooth manifold M is a map M ∋ x �→ G x such that (i) each G x is a strictly positive definite symmetric bilinear form on E ( x ), (ii) whenever V, W are smooth sections on E over an open subset Ω of M , it follows that the function Ω ∋ x �→ G x ( V ( x ) , W ( x )) is smooth. If M and E are real analytic then the metric G is real analytic if (ii’) whenever V, W are real analytic sections on E over an open subset Ω of M , it follows that the function Ω ∋ x �→ G x ( V ( x ) , W ( x )) is real analytic. 6

A WELL-KNOWN FACT: If E is a smooth (or real analytic) dis- tribution on the smooth (or real analytic) manifold M and G is a smooth (or real analytic) Riemannian metric on E , then G is the restriction to E of a smooth (or real analytic) Riemannian metric on M (that is, a Riemannian metric on the full tangent bundle TM ). A TRIVIAL WELL-KNOWN FACT: If E is a smooth (or real ana- lytic) distribution on M with fiber dimension d , and G is a smooth (or real analytic) Riemannian metric on E , then for every x ∈ M there exists an orthonormal local basis of smooth (or real analytic) sections of E near x , that is, a d -tuple ( X 1 , . . . , X d ) of smooth (or real analytic) sections of E defined on an open neighborhood Ω of x such that, for every y ∈ Ω, ( X 1 ( y ) , . . . , X d ( y )) is a basis of E ( y ) for which G y ( X i ( y ) , X j ( y )) = δ ij . 7

BRACKET-GENERATING DISTRIBUTIONS If L is any linear space of smooth vector field on a manifold M , and x ∈ M , we define L ( x ) def = { V ( x ) : V ∈ L } . We say that L has full rank at x if L ( x ) = T x M . If S is any set of smooth vector fields on M , then L [ S ] will denote the Lie algebra of vector fields generated by S , that is, the smallest Lie algebra (over R ) of vector fields that contains S . A smooth distribution E of fiber dimension d on a manifold M is bracket-generating if one of the following equivalent conditions hold: (1) L [Γ ∞ ( E, M )] has full rank at every x ∈ M . (2) For every x ∈ M , if ( X 1 , . . . , X d ) is any basis of sections of E defined on a neighborhood Ω of x , then L [ X 1 , . . . , X d ] has full rank at x . 8

SUBRIEMANNIAN MANIFOLDS A smooth subriemannian manifold is a triple ( M, E, G ) such that (1) M is a smooth manifold, (2) E is a smooth vector distribution on M , (3) G is a smooth Riemannian metric on E . Naturally, we call ( M, E, G ) real analytic if M , E and G are real analytic. 9

ADMISSIBLE ARCS If ( M, E, G ) is a subriemannian manifold, an admissible arc is an absolutely continuous arc ξ : [ a, b ] �→ M , defined on some compact interval [ a, b ], such that ˙ ξ ( t ) ∈ E ( ξ ( t )) for almost every t ∈ [ a, b ] . If we define the G -length � v � G of a tangent vector v ∈ T x M by � � v � G v ∈ E ( x ) , = G x ( v, v ) if � v � G = + ∞ if v / ∈ E ( x ) , and let the G -length of an arbitrary absolutely continuous arc ξ : [ a, b ] �→ M be the (finite or infinite) number � b def a � ˙ � ξ � G = ξ ( t ) � G dt , then it is easy to see that an absolutely continuous arc ξ : [ a, b ] �→ M is admissible if and � ξ � G < ∞ . only if 10

THE SUBRIEMANNIAN DISTANCE If M = ( M, E, G ) is a subriemannian manifold, and ξ : [ a, b ] �→ M is an absolutely continuous arc in M , we use ∂ξ to denote the ordered pair ( ξ ( a ) , ξ ( b )), and refer to ∂ξ as the endpoint value, or boundary value, of ξ . We then define the distance d M ( x, y ) between two points x, y of M to be the infimum of � ξ � G , taken over all absolutely continuous arcs ξ in M such that ∂ξ = ( x, y ). A WELL-KNOWN FACT: If M is connected, then ( M, d M ) is a metric space, whose topology is the same as the manifold topology of M . (So, in particular, d M is continuous on M × M .) (Reason: The bracket-generating condition and Chow’s Theorem imply that d M ( x, y ) < ∞ for all x, y .) 11

MINIMIZERS AND PAL MINIMIZERS An absolutely continuous arc ξ : [ a, b ] �→ M is a length minimizer if � ξ � G = d M ( ∂ξ ). An admissible arc ξ : [ a, b ] �→ M is parametrized by arc-length, or a PAL arc, if � ˙ ξ ( t ) � G = 1 for almost all t ∈ [ a, b ]. TRIVIAL FACT: Every admissible arc can be reparametrized to be- come a PAL arc. (Reason: Just use arc-length as the new time parameter.) 12

REMARK: In the real analytic case, for the purpose of the results considered here, we may always assume, without loss of generality, that the bracket-generating condition holds even if it originally does not hold. The reason is: through every point of M there passes a unique maxi- mal integral submanifold of the Lie algebra of vector fields generated by the real analytic global sections of E . And, in addition: all the results of this talk are about regularity of distance-minimizing arcs, and every such arc is entirely contained in an integral manifold, so in order to study such arcs we can restrict ourselves to an integral manifold. 13

AN IMPORTANT OPEN QUESTION Are all PAL minimizers smooth? This is well known to be true in the Riemannian case (i.e. when E = TM ), because in that case the PAL minimizers satisfy (in coordinates) the geodesic equation ξ i ( t ) + Γ i ξ j ( t ) ˙ ξ k ( t ) = 0 , ¨ jk ( ξ ( t )) ˙ � j,k where the Γ i jk are the Christoffel symbols, which are smooth functions on M . 14

THE SMOOTHNEESS PROBLEM To study the smoothness problem, it suffices to work with sufficiently short arcs, so we are allowed to assume (*) M is an open subset of R n , and E has an orthonormal basis ( f 1 , f 2 , . . . , f d ) of smooth (or real analytic) sections. It follows that the admissible arcs are exactly the arcs ξ that are trajectories of the control system d � x = ˙ u i f i ( x ) . (1) i =1 15

Precisely, an arc ξ : [ a, b ] �→ M is admissible if and only if it is absolutely continuous and satisfies d ˙ � ξ ( t ) = η i ( t ) f i ( ξ ( t )) for a . e . t , (2) i =1 for some integrable function η = ( η 1 , . . . , η d ) : [ a, b ] �→ R d . i =1 η i ( t ) 2 = 1 for a.e. t . Furthermore, ξ is PAL if and only if � d And, for a PAL arc ξ : [ a, b ] �→ M , the length of ξ is exactly b − a . So the PAL minimizers are the minimum time arcs for the control i =1 u i ( t ) 2 = 1. system (1) with contol constraint � d For convenience, we use instead the control constraint d u i ( t ) 2 ≤ 1 . � (3) i =1 (This does not change the minimizers.)