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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 $\left[0,T\right]$ and taking their values in ${ℝ}^{n}$. Consider the optimal control problem: minimize the value ${\int }_{0}^{T}{\Sigma }{u}_{i}\left(t\right)dt$ for $u\in 𝒰$, subject to the constraints: $\stackrel{˙}{q}\left(t\right)={\sum }_{i=1}^{m}{u}_{i}\left(t\right){F}_{i}\left(q\left(t\right)\right)$, $q\in U$, where $\left\{{F}_{1},\cdots ,{F}_{m}\right)$ are $m$ linearly independent vector fields generating a distribution $D$ in an open set $U$ in ${ℝ}^{n}$. The length of a curve $q$ of the above equation on $\left[0,T\right]$ and associated to $u\in U$ is given by $L\left(q\right)={\int }_{0}^{T}{\left({\sum }_{i=1}^{m}{u}_{i}^{2}\left(t\right)\right)}^{1/2}dt$. We can consider a sub-Riemannian (SR) manifold $\left(U,D,g\right)$, where $g$ is defined on $D$ by taking the ${F}_{i}$’s as orthonormal vector fields on $U$. The SR-distance between ${q}_{0},{q}_{1}\in U$ is the minimum of the length of the curves $q$ joining ${q}_{0}$ to ${q}_{i}$ and the sphere $S\left({q}_{0},r\right)$ 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:
 53C17 Sub-Riemannian geometry 49J15 Optimal control problems with ODE (existence)