Hostname: page-component-84b7d79bbc-dwq4g Total loading time: 0 Render date: 2024-07-31T10:09:50.196Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics of accessibility sets along anabnormal trajectory

Published online by Cambridge University Press:  15 August 2002

Emmanuel Trélat
Emmanuel Trélat*
Affiliation:
Université de Bourgogne, Laboratoire de Topologie, UMR 5584 du CNRS, BP. 47870, 21078 Dijon Cedex, France; trelat@topolog.u-bourgogne.fr.
Get access

Abstract

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 L2-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.

Keywords

Accessibility setabnormal trajectoryend-point mappingsingle-input affine control systemsub-Riemannian geometry.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 387 - 414
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A. Agrachev, Compactness for sub-Riemannian length minimizers and subanalyticity. Rend. Sem. Mat. Torino 56 (1998).
Agrachev, A., Quadratic mappings in geometric control theory. J. Soviet Math. 51 (1990) 2667-2734. CrossRef
A. Agrachev, Any smooth simple H1-local length minimizer in the Carnot-Caratheodory space is a C0-local length minimizer, Preprint. Labo. de Topologie, Dijon (1996).
Agrachev, A. and Sarychev, A.V., Strong minimality of abnormal geodesics for 2-distributions. J. Dynam. Control Systems 1 (1995) 139-176. CrossRef
Agrachev, A. and Sarychev, A.V., Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. H. Poincaré 13 (1996) 635-690. CrossRef
Agrachev, A. and Sarychev, A.V., On abnormal extremals for Lagrange variational problems. J. Math. Systems Estim. Control 8 (1998) 87-118.
G.A. Bliss, Lectures on the calculus of variations. U. of Chicago Press (1946).
B. Bonnard and M. Chyba, The role of singular trajectories in control theory. Springer Verlag, Math. Monograph (to be published).
Bonnard, B. and Kupka, I., 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. CrossRef
Bonnard, B. and Kupka, I., Generic properties of singular trajectories. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 167-186. CrossRef
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).
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).
H. Brezis, Analyse fonctionnelle. Masson (1993).
Bryant, R.L. and Hsu, L., Rigidity of integral curves of rank 2 distributions. Invent. Math. 114 (1993) 435-461. CrossRef
Hestenes, M.R., Applications of the theory of quadratic forms in Hilbert space to the calculus of variations. Pacific J. Math. 1 (1951) 525-581. CrossRef
E.B. Lee and L. Markus, Foundations of optimal control theory. John Wiley, New York (1967).
C. Lesiak and A.J. Krener, The existence and Uniqueness of Volterra Series for Nonlinear Systems. IEEE Trans. Automat. Control AC 23 (1978).
W.S. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics of rank two distributions. Mem. Amer. Math. Soc. 118 (1995).
Montgomery, R., Abnormal minimizers. SIAM J. Control Optim. 32 (1997) 1605-1620. CrossRef
M.A. Naimark, Linear differential operators. Frederick U. Pub. Co (1967).
L. Pontryagin et al., Théorie mathématique des processus optimaux. Eds Mir, Moscou (1974).
A.V. Sarychev, The index of the second variation of a control system. Math. USSR Sbornik 41 (1982).
Trélat, E., 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. CrossRef
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).
Zhong, Ge, Horizontal path space and Carnot-Caratheodory metric. Pacific J. Math. 161 (1993) 255-286.