zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the role of abnormal minimizers in sub-Riemannian geometry. (English) Zbl 1017.53034

Let 𝒰 be an open set of bounded measurable mappings u defined on [0,T] and taking their values in n . Consider the optimal control problem: minimize the value 0 T Σu i (t)dt for u𝒰, subject to the constraints: q ˙(t)= i=1 m u i (t)F i (q(t)), qU, where {F 1 ,,F m ) are m linearly independent vector fields generating a distribution D in an open set U in n . The length of a curve q of the above equation on [0,T] and associated to uU is given by L(q)= 0 T ( i=1 m u i 2 (t)) 1/2 dt. We can consider a sub-Riemannian (SR) manifold (U,D,g), where g is defined on D by taking the F i ’s as orthonormal vector fields on U. The SR-distance between q 0 ,q 1 U is the minimum of the length of the curves q joining q 0 to q i and the sphere S(q 0 ,r) with radius r is defined.

The authors give a geometric framework to analyse the singularities of the sphere in the abnormal directions and compute asymptotics of the distance in those directions, mainly in the Martinet case. After recalling the Hamiltonian formalism and the generalities concerning SR geometry the authors analyse the role of abnormal geodesics in SR Martinet geometry and study to which category the sphere belongs. Finally after defining the Martinet sector, the authors describe a Martinet sector in the n-dimensional SR-sphere using the computations in the previous sections by means of the Hamiltonian formalism and microlocal analysis.

MSC:
53C17Sub-Riemannian geometry
49J15Optimal control problems with ODE (existence)