Asymptotics of accessibility sets along an abnormal trajectory
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 387-414.

We describe precisely, under generic conditions, the contact of the accessibility set at time T with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-riemannian system of rank 2. As a consequence we obtain in sub-riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin’s cone along γ, called the L ∞ -sector and the L 2 -sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.

Classification : 93B03, 49K15
Mots clés : accessibility set, abnormal trajectory, end-point mapping, single-input affine control system, sub-riemannian geometry 
@article{COCV_2001__6__387_0,
     author = {Tr\'elat, Emmanuel},
     title = {Asymptotics of accessibility sets along an abnormal trajectory},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {387--414},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
     mrnumber = {1836049},
     zbl = {0996.93009},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2001__6__387_0/}
}
TY  - JOUR
AU  - Trélat, Emmanuel
TI  - Asymptotics of accessibility sets along an abnormal trajectory
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2001
SP  - 387
EP  - 414
VL  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/COCV_2001__6__387_0/
LA  - en
ID  - COCV_2001__6__387_0
ER  - 
%0 Journal Article
%A Trélat, Emmanuel
%T Asymptotics of accessibility sets along an abnormal trajectory
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2001
%P 387-414
%V 6
%I EDP-Sciences
%U http://www.numdam.org/item/COCV_2001__6__387_0/
%G en
%F COCV_2001__6__387_0
Trélat, Emmanuel. Asymptotics of accessibility sets along an abnormal trajectory. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 387-414. http://www.numdam.org/item/COCV_2001__6__387_0/

[1] A. Agrachev, Compactness for sub-Riemannian length minimizers and subanalyticity. Rend. Sem. Mat. Torino 56 (1998). | MR | Zbl

[2] A. Agrachev, Quadratic mappings in geometric control theory. J. Soviet Math. 51 (1990) 2667-2734.

[3] A. Agrachev, Any smooth simple H 1 -local length minimizer in the Carnot-Caratheodory space is a C 0 -local length minimizer, Preprint. Labo. de Topologie, Dijon (1996).

[4] A. Agrachev and A.V. Sarychev, Strong minimality of abnormal geodesics for 2-distributions. J. Dynam. Control Systems 1 (1995) 139-176. | MR | Zbl

[5] A. Agrachev and A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. H. Poincaré 13 (1996) 635-690. | Numdam | MR | Zbl

[6] A. Agrachev and A.V. Sarychev, On abnormal extremals for Lagrange variational problems. J. Math. Systems Estim. Control 8 (1998) 87-118. | MR | Zbl

[7] G.A. Bliss, Lectures on the calculus of variations. U. of Chicago Press (1946). | MR | Zbl

[8] B. Bonnard and M. Chyba, The role of singular trajectories in control theory. Springer Verlag, Math. Monograph (to be published).

[9] B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal. Forum Math. 5 (1993) 111-159. | Zbl

[10] B. Bonnard and I. Kupka, Generic properties of singular trajectories. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 167-186. | Numdam | MR | Zbl

[11] B. Bonnard and E. Trélat, On the role of abnormal minimizers in SR-geometry, Preprint. Labo. Topologie Dijon. Ann. Fac. Sci. Toulouse (to be published).

[12] B. Bonnard and E. Trélat, Stratification du secteur anormal dans la sphère de Martinet de petit rayon, edited by A. Isidori, F. Lamnabhi Lagarrigue and W. Respondek. Springer, Lecture Notes in Control and Inform. Sci. 259, Nonlinear Control in the Year 2000, Vol. 2. Springer (2000). | MR

[13] H. Brezis, Analyse fonctionnelle. Masson (1993). | MR | Zbl

[14] R.L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions. Invent. Math. 114 (1993) 435-461. | MR | Zbl

[15] M.R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations. Pacific J. Math. 1 (1951) 525-581. | MR | Zbl

[16] E.B. Lee and L. Markus, Foundations of optimal control theory. John Wiley, New York (1967). | MR | Zbl

[17] C. Lesiak and A.J. Krener, The existence and Uniqueness of Volterra Series for Nonlinear Systems. IEEE Trans. Automat. Control AC 23 (1978). | MR | Zbl

[18] W.S. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics of rank two distributions. Mem. Amer. Math. Soc. 118 (1995). | MR | Zbl

[19] R. Montgomery, Abnormal minimizers. SIAM J. Control Optim. 32 (1997) 1605-1620. | MR | Zbl

[20] M.A. Naimark, Linear differential operators. Frederick U. Pub. Co (1967). | MR

[21] L. Pontryagin et al., Théorie mathématique des processus optimaux. Eds Mir, Moscou (1974). | MR

[22] A.V. Sarychev, The index of the second variation of a control system. Math. USSR Sbornik 41 (1982). | Zbl

[23] E. Trélat, Some properties of the value function and its level sets for affine control systems with quadratic cost. J. Dynam. Control Systems 6 (2000) 511-541. | MR | Zbl

[24] E. Trélat, Étude asymptotique et transcendance de la fonction valeur en contrôle optimal ; catégorie log-exp dans le cas sous-Riemannien de Martinet, Ph.D. Thesis. Université de Bourgogne, Dijon, France (2000).

[25] Zhong Ge, Horizontal path space and Carnot-Caratheodory metric. Pacific J. Math. 161 (1993) 255-286. | MR | Zbl