Gradient horizontal de fonctions polynomiales  [ Horizontal gradient of polynomial functions ]
Annales de l'Institut Fourier, Volume 59 (2009) no. 5, p. 1999-2042

We study trajectories of sub-Riemannian (also called horizontal) gradient of polynomials. In this setting Łojasiewicz’s gradient inequality does not hold and a trajectory of a horizontal gradient may be of infinite length, moreover it may accumulate on a closed curve. We show that these phenomena are exceptional; for a generic polynomial function the behavior of the trajectories of horizontal gradients are similar to the behavior of the trajectories of a Riemannian gradient. To obtain the finiteness of the length of trajectories we change suitably the sub-Riemannian metric. We consider a class of splitting distributions which contains those of Heisenberg and Martinet. For a generic polynomial f the set V f of horizontal critical points, is a smooth algebraic set of dimension 1 or the empty set, moreover f| V f is a Morse function. We show that for a generic polynomial function any trajectory of the horizontal gradient (which approaches V f ) has a limit, as in the Riemannian case studied by S. Łojasiewicz.

Nous étudions les trajectoires du gradient sous-riemannien (appellé horizontal) de fonctions polynômes. Dans ce cadre l’inégalité de Łojasiewicz n’est pas valide et une trajectoire du gradient horizontal peut être de longueur infinie, et peut même s’accumuler sur une courbe fermée. Nous montrons que ces comportement sont exceptionnels ; et que, pour une fonction générique les trajectoires de son gradient horizontal ont des propriétés similaires au cas du gradient riemannien. Pour obtenir la finitude des longueurs des trajectoires, nous changeons la métrique sous-riemanienne de façon convenable. Nous considérons une classe de distributions dites scindées, incluant celles d’Heisenberg et de Martinet. Pour un polynôme générique f l’ensemble V f des points critiques horizontaux de f est un ensemble algébrique lisse de dimension 1 ou est vide et la restriction f| V f est une fonction de Morse. Nous montrons aussi que pour un polynôme générique f, chaque trajectoire du gradient horizontal (qui approche V f ) possède une limite comme dans le cas riemannien étudié par S. Łojasiewicz.

DOI : https://doi.org/10.5802/aif.2481
Classification:  14P10,  53C17,  58Kxx,  58A30,  58K14,  93F14
Keywords: Sub-Riemannian, gradient, inegality
@article{AIF_2009__59_5_1999_0,
     author = {Dinh, Si Tiep and Kurdyka, Krzysztof and Orro, Patrice},
     title = {Gradient horizontal de fonctions polynomiales},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {5},
     year = {2009},
     pages = {1999-2042},
     doi = {10.5802/aif.2481},
     mrnumber = {2573195},
     zbl = {1197.14058},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2009__59_5_1999_0}
}
Dinh, Si Tiep; Kurdyka, Krzysztof; Orro, Patrice. Gradient horizontal de fonctions polynomiales. Annales de l'Institut Fourier, Volume 59 (2009) no. 5, pp. 1999-2042. doi : 10.5802/aif.2481. http://www.numdam.org/item/AIF_2009__59_5_1999_0/

[1] Absil, P.-A.; Kurdyka, K. On the stable equilibrium points of gradient systems, Systems Control Lett., Tome 55 (2006) no. 7, pp. 573-577 | Article | MR 2225367 | Zbl 1129.34320

[2] Baeg, M.; Helmke, U.; Moore, J. B. Gradient flow techniques for pose estimation of quadratic surfaces, Proceedings of the World Congress in Computational Methods and Applied Mathematics (1994)

[3] Balogh, Z. M.; Holopainen, I.; Tyson, J. T. Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups, Math. Ann., Tome 324 (2002), pp. 159-186 | Article | MR 1931762 | Zbl 1014.22009

[4] Benedetti, R.; Risler, J-J. Real algebraic and semi-algebraic sets, Hermann (1991) | MR 1070358 | Zbl 0694.14006

[5] Bierstone, E.; Milman, P. D. Semianalytic and subanalytic sets, Inst. Hautes Etudes Sci. Publ. Math. (1988) no. 67, pp. 5-42 | Article | Numdam | MR 972342 | Zbl 0674.32002

[6] Bochnak, J.; Coste, M.; Roy, M-F. Géométrie semi-algébrique réelle, Springer (1987) | MR 949442

[7] Chill, R. The Lojasiewicz-Simon gradient inequality, J. Funct. Anal., Tome 201 (2003), pp. 572-601 | Article | MR 1986700 | Zbl 1036.26015

[8] Chow, W. L. Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., Tome 117 (1939), pp. 98-105 | Article | MR 1880

[9] D. D’Acunto, K. Kurdyka Bounds for gradient trajectories of polynomial and definable functions with application ((soumis) J. Diff. Geometry (2004))

[10] D’Acunto, D.; Kurdyka, K. Explicit bounds for the Lojasiewicz exponent in the gradient inequality for polynomials, Ann. Polon. Math., Tome 87 (2005), pp. 51-61 | Article | MR 2208535 | Zbl 1093.32011

[11] Gabrielov, A. Multiplicities of Pffafian intersections and the Lojasiewicz inequality, Selecta Math. (N.S.), Tome 1 (1995), pp. 113-127 | Article | MR 1327229 | Zbl 0889.32005

[12] Goresky, M.; Macpherson, R. Stratified Morse theory, Springer (1988) | MR 932724 | Zbl 0639.14012

[13] Gromov, M. Carnot-Caratheodory spaces seen from within. Subriemannian Geometry, Birkhäuser Verlag, Progress in Mathematics, Tome 144 (1996) | MR 1421823 | Zbl 0864.53025

[14] Guillemin, V.; Pollack, A. Differential topology, Prentice-Hall (1974) | MR 348781 | Zbl 0361.57001

[15] Helmke, U.; Moore, J. B. Optimization and dynamical systems, Springer (1994) | MR 1299725 | Zbl 0943.93001

[16] Hirsch, M. Differential topology, Springer (1976) | MR 448362 | Zbl 0356.57001

[17] Huang, S.-Z. Gradient inequalities with applications to asymptotic behavior and stability of gradient-like systems, AMS Mathematical Surveys and Monographs Tome 126 (2006) | MR 2226672 | Zbl 1132.35002

[18] Jiang, D.; Moore, J. B. A gradient flow approach to decentralised output feedback optimal control, Systems Control Lett., Tome 27 (1996) no. 4, pp. 223-231 | Article | MR 1389555 | Zbl 0875.93020

[19] Kurdyka, K. On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier (Grenoble), Tome 48 (1998) no. 3, pp. 769-783 | Article | Numdam | MR 1644089 | Zbl 0934.32009

[20] Kurdyka, K.; Mostowski, T.; Parusiński, A. Proof of the Gradient Conjecture of R. Thom, Ann. of Math., Tome 152 (2000), pp. 163-792 | Article | MR 1815701 | Zbl 1053.37008

[21] Kurdyka, K.; Parusiński, A. w f -stratification of subanalytic functions and the Łojasiewicz inequality, C. R. Acad. Sci. Paris Sér. I Math. , Tome 318 (1994) no. 2, pp. 129-133 | Zbl 0799.32007

[22] Łojasiewicz, S.; B. Malgrange (Paris 1962). Publications Du Cnrs Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du CNRS. Les équations aux dérivées partielles, Paris, Tome 117 (1963) | Zbl 0234.57007

[23] Łojasiewicz, S. Ensembles semi-analytiques, I.H.E.S. Bures-sur-Yvette (1965)

[24] Łojasiewicz, S. Sur les trajectoires du gradient d’une fonction analytique, Seminari di Geometria, Bologna (1982-1983), pp. 115-117 | MR 771152 | Zbl 0606.58045

[25] Łojasiewicz, S. Sur la géométrie semi- et sous- analytique, Ann. Inst. Fourier (Grenoble), Tome 43 (1993) no. 5, pp. 1575-1595 | Article | Numdam | MR 1275210 | Zbl 0803.32002

[26] Lu, Y. C. Singularity theory and an introduction to catastrophe theory, Springer (1976) | MR 461562 | Zbl 0354.58008

[27] Magnani, V. A Blow-up theorem for regular hypersurfaces on nilpotent groups, Manuscripta Math., Tome 110 (2003) no. 1, pp. 55-76 | Article | MR 1951800 | Zbl 1010.22010

[28] Manton, J. H.; Helmke, U.; Mareels, I. M. Y. A dual purpose principal and minor component flow, Systems Control Lett., Tome 54 (2005) no. 8, pp. 759-769 | Article | MR 2147235 | Zbl 1129.34319

[29] Rashevsky, P. K. Any two points of a totally nonholonomic space may be connected by an admissible line, Uch. Zap. Ped. Inst. im. Liebknechta. Ser. Phys. Math., Tome 2 (1938), pp. 83-94

[30] Ridout, D.; Judd, K. Convergence properties of gradient descent noise reduction, Physica. D, Tome 165 (2002) no. 1-2, pp. 26-47 | Article | MR 1910616 | Zbl 1008.37049

[31] Simon, L. Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2), Tome 118 (1983) no. 3, pp. 525-571 | Article | MR 727703 | Zbl 0549.35071

[32] Strichartz, R. S. Sub-riemannian geometry, J. Diff. Geom., Tome 24 (1986), pp. 221-263 | MR 862049 | Zbl 0609.53021

[33] Sussmann, H. J. Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., Tome 180 (1973), pp. 171-188 | Article | MR 321133 | Zbl 0274.58002

[34] Thom, R. Problèmes rencontrés dans mon parcours mathématiques : un bilan, Publ. Math. IHES, Tome 70 (1989), pp. 200-214 | Numdam | MR 1067383 | Zbl 0709.58001

[35] Trendafilov, N. T.; Lippert, R. A. The multimode Procrustes problem, Linear Algebra Appl., Tome 349 (2002), pp. 245-264 | Article | MR 1903736 | Zbl 0999.65051

[36] Yan, W. Y.; Teo, K. L.; Moore, J. B. A gradient flow approach to computing LQ optimal output feedback gains, Optimal Control Appl. Methods, Tome 15 (1994) no. 1, pp. 67-75 | Article | MR 1263418 | Zbl 0815.49025

[37] Yoshizawa, S.; Helmke, U.; Starkow, K. Convergence analysis for principal component flows, Mathematical theory of networks and systems (Perpignan, 2000). Int. J. Appl. Math. Comput. Sci., Tome 11 (2001) no. 1, pp. 223-236 | MR 1835155 | Zbl pre01599128