Gradient horizontal de fonctions polynomiales

Dinh, Si Tiep; Kurdyka, Krzysztof; Orro, Patrice

Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1999-2042.

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.

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.

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

[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., Volume 324 (2002), pp. 159-186 | DOI | MR | Zbl

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

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

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

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

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

[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., Volume 87 (2005), pp. 51-61 | DOI | MR | Zbl

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

[12] Goresky, M.; MacPherson, R. Stratified Morse theory, Springer, 1988 | MR | Zbl

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

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

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

[16] Hirsch, M. Differential topology, Springer, 1976 | MR | Zbl

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

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

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

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

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

[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, Volume 117, Paris (1963) | Zbl

[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 | Zbl

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

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

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

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

[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., Volume 2 (1938), pp. 83-94

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

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

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

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

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

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

[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, Volume 15 (1994) no. 1, pp. 67-75 | DOI | MR | Zbl

[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., Volume 11 (2001) no. 1, pp. 223-236 | MR