On the stabilization problem for nonholonomic distributions

Journal of the European Mathematical Society

Volumn 11, Issue 2, 2009, Pages 223-255

  Rifford, L ^a   Trelat E ^b  

^a UNIVERSITÉ CÔTE D'AZUR   (France)

^b UNIVERSITÉ D'ORLÉANS   (France)

Author keywords

Martinet case; Minimizing singular path; Nonholonomic distributions; Nonsmooth analysis; SRS feedback; Stabilization

Indexed keywords

EID: 74249110334     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/148     Document Type: Article

References (48)

1. Agrachev, A.: Compactness for sub-Riemannian length-minimizers and subanalyticity. In: Control Theory and its Applications (Grado, 1998), Rend. Sem. Mat. Univ. Politec. Torino 56, 1-12 (2001) Zbl 1039.53038 MR 1845741
2. Agrachev, A., Bonnard, B., Chyba, M., Kupka, I.: Sub-Riemannian sphere in Martinet flat case. ESAIM Control Optim. Calc. Var. 2, 377-448 (1997) Zbl 0902.53033 MR 1483765
3. Agrachev, A., Sachkov, Y.: Control Theory from the Geometric Viewpoint. Encyclopaedia Math. Sci. 87, Control Theory and Optimization, II, Springer, Berlin (2004) Zbl 1062.93001 MR 2062547
4. Agrachev, A., Sarychev, A.: Sub-Riemannian metrics: minimality of abnormal geodesics versus subanalyticity. ESAIM Control Optim. Calc. Var. 4, 377-403 (1999) Zbl 0978.53065 MR 1693912
5. Alberti, G., Ambrosio, L., Cannarsa, P.: On the singularities of convex functions. Manuscripta Math. 76, 421-435 (1992) Zbl 0784.49011 MR 1185029
6. Ancona, F., Bressan, A.: Patchy vector fields and asymptotic stabilization. ESAIM Control Optim. Calc. Var. 4, 445-471 (1999) Zbl 0924.34058 MR 1693900
7. Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi. Math. Appl. 17, Springer, Berlin (1994) Zbl 0819.35002 MR 1613876
8. Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton- Jacobi-Bellman equations. Systems & Control Found. Appl., Birkhäuser Boston, Boston, MA (1997) Zbl 0890.49011 MR 1484411
9. Bellaïche, A.: The tangent space in sub-Riemannian geometry. In: Sub-Riemannian Geometry, Birkhäuser, 1-78 (1996) Zbl 0862.53031 MR 1421822
10. Bismut, J.-M.: Large Deviations and the Malliavin Calculus. Progr. Math. 45, Birkhäuser (1984) Zbl 0537.35003 MR 0755001
11. Bonnard, B., Trélat, E.: On the role of abnormal minimizers in sub-Riemannian geometry. Ann. Fac. Sci. Toulouse Math. (6) 10, 405-491 (2001) Zbl 1017.53034 MR 1923686
12. Bredon, G. E.: Topology and Geometry. Grad. Texts in Math. 139, Springer, New York (1993) Zbl 0934.55001 MR 1700700
13. Brockett, R. W.: Asymptotic stability and feedback stabilization. In: Differential Geometric Control Theory, R. W. Brockett et al. (eds.), Birkhäuser, Boston, 181-191 (1983) Zbl 0528.93051 MR 0708502
14. Cannarsa, P., Rifford, L.: Semiconcavity results for optimal control problems admitting no singular minimizing controls. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 773-802 (2008) Zbl 1145.49022 MR 2436793
15. Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control. Progr. Nonlinear Differential Equations Appl. 58, Birkhäuser Boston, Boston, MA (2004) Zbl 1095.49003 MR 2041617
16. Chitour, Y., Jean, F., Trélat, E.: Propriétés génériques des trajectoires singulières. C. R. Acad. Sci. Paris Sér. I Math. 337, 49-52 (2003) Zbl 1038.58019 MR 1986538
17. Chitour, Y., Jean, F., Trélat, Genericity results for singular curves. J. Differential Geom. 73, E., 45-73 (2006) Zbl 1102.53019 MR 2217519
18. Chitour, Y., Jean, F., Trélat, E.: Singular trajectories of control-affine systems. SIAM J. Control Optim. 47, 1078-1095 (2008) MR 2385874
19. Chow, W. L.: Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117, 98-105 (1939) Zbl 0022.02304 MR 0001880
20. Clarke, F. H.: Optimization and Nonsmooth Analysis. 2nd ed., Classics in Appl. Math. 5, SIAM, Philadelphia (1990) Zbl 0696.49002 MR 1058436
21. Clarke, F. H., Ledyaev, Yu. S., Sontag, E. D., Subbotin, A. I.: Asymptotic controllabilityimplies feedback stabilization. I.E.E.E. Trans. Automatic Control 42, 1394-1407 (1997) Zbl 0892.93053 MR 1472857
22. Clarke, F. H., Ledyaev, Yu. S., Stern, R. J., Wolenski, P. R.: Nonsmooth Analysis and Control Theory. Grad. Texts in Math. 178, Springer, New York (1998) Zbl 1047.49500 MR 1488695
23. Coron, J.-M.: A necessary condition for feedback stabilization. Systems Control Lett. 14, 227-232 (1994) Zbl 0699.93075 MR 1049357
24. Coron, J.-M.: Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5, 295-312 (1992) Zbl 0760.93067 MR 1164379
25. Crandall, M. G., Lions, P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277, 1-42 (1983) Zbl 0599.35024 MR 0690039
26. Federer, H.: Geometric Measure Theory. Grundlehren Math. Wiss. 153, Springer, New York (1969) Zbl 0176.00801 MR 0257325
27. Hardt, R. M.: Stratification of real analytic mappings and images. Invent. Math. 28, 193-208 (1975) Zbl 0298.32003 MR 0372237
28. Hironaka, H.: Subanalytic sets. In: Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo, 453-493 (1973) Zbl 0297.32008 MR 0377101
29. Hsu, L.: Calculus of variations via the Griffiths formalism. J. Differential Geom. 36, 551-589 (1992) Zbl 0768.49014 MR 1189496
30. Li, Y. Y., Nirenberg, L.: The distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Comm. Pure Appl. Math. 58, 85-146 (2005) Zbl 1062.49021 MR 2094267
31. Lions, P.-L.: Generalized Solutions of Hamilton-Jacobi Equations. Pitman, Boston, MA (1982) Zbl 0497.35001 MR 0667669
32. Liu, W., J. Sussmann, H.: Shortest paths for sub-Riemannian metrics on rank-2 distributions. Mem. Amer. Math. Soc. 118, no.564 (1995) Zbl 0843.53038 MR 1303093
33. Montgomery, R.: A Tour of Subriemannian Geometries, their Geodesics and Applications. Math. Surveys Monogr. 91, Amer. Math. Soc., Providence, RI (2002) Zbl 1044.53022 MR 1867362
34. Orro, P., Pelletier, F.: Differential properties of the distance function associated to a submanifold in sub-Riemannian geometry. Prépublication du LAMA (2001)
35. Pontryagin, L., Boltyanskiǐ, V., Gamkrelidze, R., Mishchenko, E.: The Mathematical Theory of Optimal Processes. Wiley Interscience (1962) Zbl 0882.01027 MR 0166036
36. Rashevskiǐ, P. K.: About connecting two points of a completely nonholonomic space by admissible curve. Uch. Zapiski Ped. Inst. Libknechta 2, 83-94 (1938)
37. Rifford, L.: Semiconcave control-Lyapunov functions and stabilizing feedbacks. SIAM J. Control Optim. 41, 659-681 (2002) Zbl 1034.93053 MR 1939865
38. Rifford, L.: Stratified semiconcave control-Lyapunov functions and the stabilization problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 343-384 (2005) Zbl 1114.93081 MR 2136728
39. Rifford, L.: The stabilization problem: AGAS and SRS feedbacks. In: Optimal Control, Stabilization, and Nonsmooth Analysis, Lecture Notes in Control Inform. Sci. 301, Springer, Heidelberg, 173-184 (2004) Zbl pre02222962 MR 2079683
40. Rifford, L.: The stabilization problem on surfaces. Rend. Sem. Mat. Univ. Politec. Torino 64, 55-61 (2006) MR 2201073
41. Rifford, L.: On the existence of local smooth repulsive stabilizing feedbacks in dimension three. J. Differential Equations 226, 429-500 (2006) Zbl pre05045577 MR 2237687
42. Rifford, L.: Nonholonomic Variations. Monograph, in progress.
43. Rifford, L., Trélat, E.: Morse-Sard type results in sub-Riemannian geometry. Math. Ann. 332, 145-159 (2005) Zbl 1069.53033 MR 2139255
44. Ryan, E. P.: On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J. Control Optim. 32, 1597-1604 (1994) Zbl 0806.93049 MR 1297100
45. Sarychev, A.: The index of the second variation of a control system. Math. USSR-Sb. 41, 383-401 (1982) Zbl 0484.49012
46. Sussmann, H. J.: Subanalytic sets and feedback control. J. Differential Equations 31, 31-52 (1979) Zbl 0407.93010 MR 0524816
47. Trélat, E.: Some properties of the value function and its level sets for affine control systems with quadratic cost. J. Dynam. Control Systems 6, 511-541 (2000) Zbl 0964.49021 MR 1778212
48. Zhitomirskiǐ, M.: Typical Singularities of Differential 1-Forms and Pfaffian Equations. Transl. Math. Monogr. 113, Amer. Math. Soc. (1992) MR 1195792