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Si Tiep Dinh; Krzysztof Kurdyka; Patrice Orro
Gradient horizontal de fonctions polynomiales
(Horizontal gradient of polynomial functions)
Annales de l'institut Fourier, 59 no. 5 (2009), p. 1999-2042, doi: 10.5802/aif.2481
Article: subscription required (your ip address: 107.22.156.205) | Reviews MR 2573195 | Zbl 1197.14058
Class. Math.: 14P10, 53C17, 58Kxx, 58A30, 58K14, 93F14
Keywords: Sub-Riemannian, gradient, inegality
[1] P.-A. Absil & K. Kurdyka, “On the stable equilibrium points of gradient systems”, Systems Control Lett. 55 (2006) no. 7, p. 573-577
Article | MR 2225367 | Zbl 1129.34320
[2] M. Baeg, U. Helmke & J. B. Moore, Gradient flow techniques for pose estimation of quadratic surfaces, in Proceedings of the World Congress in Computational Methods and Applied Mathematics, 1994
[3] Z. M. Balogh, I. Holopainen & J. T. Tyson, “Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups”, Math. Ann. 324 (2002), p. 159-186
Article | MR 1931762 | Zbl 1014.22009
[4] R. Benedetti & J-J. Risler, Real algebraic and semi-algebraic sets, Hermann, 1991 MR 1070358 | Zbl 0694.14006
[5] E. Bierstone & P. D. Milman, “Semianalytic and subanalytic sets”, Inst. Hautes Etudes Sci. Publ. Math. (1988) no. 67, p. 5-42
Numdam | MR 972342 | Zbl 0674.32002
[6] J. Bochnak, M. Coste & M-F. Roy, Géométrie semi-algébrique réelle, Springer, 1987 MR 949442
[7] R. Chill, “The Lojasiewicz-Simon gradient inequality”, J. Funct. Anal. 201 (2003), p. 572-601
Article | MR 1986700 | Zbl 1036.26015
[8] W. L. Chow, “Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung”, Math. Ann. 117 (1939), p. 98-105
Article | MR 1880 | JFM 65.0398.01
[9] K. Kurdyka D. D’Acunto, “Bounds for gradient trajectories of polynomial and definable functions with application”, (soumis) J. Diff. Geometry (2004)
[10] D. D’Acunto & K. Kurdyka, “Explicit bounds for the Lojasiewicz exponent in the gradient inequality for polynomials”, Ann. Polon. Math. 87 (2005), p. 51-61
Article | MR 2208535 | Zbl 1093.32011
[11] A. Gabrielov, “Multiplicities of Pffafian intersections and the Lojasiewicz inequality”, Selecta Math. (N.S.) 1 (1995), p. 113-127
Article | MR 1327229 | Zbl 0889.32005
[12] M. Goresky & R. MacPherson, Stratified Morse theory, Springer, 1988 MR 932724 | Zbl 0639.14012
[13] M. Gromov, Carnot-Caratheodory spaces seen from within. Subriemannian Geometry, Progress in Mathematics 144, Birkhäuser Verlag, 1996 MR 1421823 | Zbl 0864.53025
[14] V. Guillemin & A. Pollack, Differential topology, Prentice-Hall, 1974 MR 348781 | Zbl 0361.57001
[15] U. Helmke & J. B. Moore, Optimization and dynamical systems, Springer, 1994 MR 1299725 | Zbl 0943.93001
[16] M. Hirsch, Differential topology, Springer, 1976 MR 448362 | Zbl 0356.57001
[17] S.-Z. Huang, Gradient inequalities with applications to asymptotic behavior and stability of gradient-like systems 126, AMS Mathematical Surveys and Monographs, 2006 MR 2226672 | Zbl 1132.35002
[18] D. Jiang & J. B. Moore, “A gradient flow approach to decentralised output feedback optimal control”, Systems Control Lett. 27 (1996) no. 4, p. 223-231
Article | MR 1389555 | Zbl 0875.93020
[19] K. Kurdyka, “On gradients of functions definable in o-minimal structures”, Ann. Inst. Fourier (Grenoble) 48 (1998) no. 3, p. 769-783
Cedram | MR 1644089 | Zbl 0934.32009
[20] K. Kurdyka, T. Mostowski & A. Parusiński, “Proof of the Gradient Conjecture of R. Thom”, Ann. of Math. 152 (2000), p. 163-792
Article | MR 1815701 | Zbl 1053.37008
[21] K. Kurdyka & A. Parusiński, “$w_f$-stratification of subanalytic functions and the Łojasiewicz inequality”, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994) no. 2, p. 129-133 Zbl 0799.32007
[22] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in B. Malgrange (Paris 1962). Publications du CNRS, ed., Colloques internationaux du CNRS. Les équations aux dérivées partielles, 1963 Zbl 0234.57007
[23] S. Łojasiewicz, Ensembles semi-analytiques, I.H.E.S. Bures-sur-Yvette, 1965
[24] S. Łojasiewicz, Sur les trajectoires du gradient d’une fonction analytique, in Seminari di Geometria, 1982-1983, p. 115-117 MR 771152 | Zbl 0606.58045
[25] S. Łojasiewicz, “Sur la géométrie semi- et sous- analytique”, Ann. Inst. Fourier (Grenoble) 43 (1993) no. 5, p. 1575-1595
Cedram | MR 1275210 | Zbl 0803.32002
[26] Y. C. Lu, Singularity theory and an introduction to catastrophe theory, Springer, 1976 MR 461562 | Zbl 0354.58008
[27] V. Magnani, “A Blow-up theorem for regular hypersurfaces on nilpotent groups”, Manuscripta Math. 110 (2003) no. 1, p. 55-76
Article | MR 1951800 | Zbl 1010.22010
[28] J. H. Manton, U. Helmke & I. M. Y. Mareels, “A dual purpose principal and minor component flow”, Systems Control Lett. 54 (2005) no. 8, p. 759-769
Article | MR 2147235 | Zbl 1129.34319
[29] P. K. Rashevsky, “Any two points of a totally nonholonomic space may be connected by an admissible line”, Uch. Zap. Ped. Inst. im. Liebknechta. Ser. Phys. Math. 2 (1938), p. 83-94
[30] D. Ridout & K. Judd, “Convergence properties of gradient descent noise reduction”, Physica. D 165 (2002) no. 1-2, p. 26-47
Article | MR 1910616 | Zbl 1008.37049
[31] L. Simon, “Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems”, Ann. of Math. (2) 118 (1983) no. 3, p. 525-571
Article | MR 727703 | Zbl 0549.35071
[32] R. S. Strichartz, “Sub-riemannian geometry”, J. Diff. Geom. 24 (1986), p. 221-263
Article | MR 862049 | Zbl 0609.53021
[33] H. J. Sussmann, “Orbits of families of vector fields and integrability of distributions”, Trans. Amer. Math. Soc. 180 (1973), p. 171-188
Article | MR 321133 | Zbl 0274.58002
[34] R. Thom, “Problèmes rencontrés dans mon parcours mathématiques : un bilan”, Publ. Math. IHES 70 (1989), p. 200-214
Numdam | MR 1067383 | Zbl 0709.58001
[35] N. T. Trendafilov & R. A. Lippert, “The multimode Procrustes problem”, Linear Algebra Appl. 349 (2002), p. 245-264
Article | MR 1903736 | Zbl 0999.65051
[36] W. Y. Yan, K. L. Teo & J. B. Moore, “A gradient flow approach to computing LQ optimal output feedback gains”, Optimal Control Appl. Methods 15 (1994) no. 1, p. 67-75
Article | MR 1263418 | Zbl 0815.49025
[37] S. Yoshizawa, U. Helmke & K. Starkow, “Convergence analysis for principal component flows”, Mathematical theory of networks and systems (Perpignan, 2000). Int. J. Appl. Math. Comput. Sci. 11 (2001) no. 1, p. 223-236 MR 1835155 | Zbl pre01599128
Si Tiep Dinh; Krzysztof Kurdyka; Patrice Orro
Gradient horizontal de fonctions polynomiales
(Horizontal gradient of polynomial functions)
Annales de l'institut Fourier, 59 no. 5 (2009), p. 1999-2042, doi: 10.5802/aif.2481
Article: subscription required (your ip address: 107.22.156.205) | Reviews MR 2573195 | Zbl 1197.14058
Class. Math.: 14P10, 53C17, 58Kxx, 58A30, 58K14, 93F14
Keywords: Sub-Riemannian, gradient, inegality
Résumé - Abstract
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.
Bibliography
Article | MR 2225367 | Zbl 1129.34320
[2] M. Baeg, U. Helmke & J. B. Moore, Gradient flow techniques for pose estimation of quadratic surfaces, in Proceedings of the World Congress in Computational Methods and Applied Mathematics, 1994
[3] Z. M. Balogh, I. Holopainen & J. T. Tyson, “Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups”, Math. Ann. 324 (2002), p. 159-186
Article | MR 1931762 | Zbl 1014.22009
[4] R. Benedetti & J-J. Risler, Real algebraic and semi-algebraic sets, Hermann, 1991 MR 1070358 | Zbl 0694.14006
[5] E. Bierstone & P. D. Milman, “Semianalytic and subanalytic sets”, Inst. Hautes Etudes Sci. Publ. Math. (1988) no. 67, p. 5-42
Numdam | MR 972342 | Zbl 0674.32002
[6] J. Bochnak, M. Coste & M-F. Roy, Géométrie semi-algébrique réelle, Springer, 1987 MR 949442
[7] R. Chill, “The Lojasiewicz-Simon gradient inequality”, J. Funct. Anal. 201 (2003), p. 572-601
Article | MR 1986700 | Zbl 1036.26015
[8] W. L. Chow, “Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung”, Math. Ann. 117 (1939), p. 98-105
Article | MR 1880 | JFM 65.0398.01
[9] K. Kurdyka D. D’Acunto, “Bounds for gradient trajectories of polynomial and definable functions with application”, (soumis) J. Diff. Geometry (2004)
[10] D. D’Acunto & K. Kurdyka, “Explicit bounds for the Lojasiewicz exponent in the gradient inequality for polynomials”, Ann. Polon. Math. 87 (2005), p. 51-61
Article | MR 2208535 | Zbl 1093.32011
[11] A. Gabrielov, “Multiplicities of Pffafian intersections and the Lojasiewicz inequality”, Selecta Math. (N.S.) 1 (1995), p. 113-127
Article | MR 1327229 | Zbl 0889.32005
[12] M. Goresky & R. MacPherson, Stratified Morse theory, Springer, 1988 MR 932724 | Zbl 0639.14012
[13] M. Gromov, Carnot-Caratheodory spaces seen from within. Subriemannian Geometry, Progress in Mathematics 144, Birkhäuser Verlag, 1996 MR 1421823 | Zbl 0864.53025
[14] V. Guillemin & A. Pollack, Differential topology, Prentice-Hall, 1974 MR 348781 | Zbl 0361.57001
[15] U. Helmke & J. B. Moore, Optimization and dynamical systems, Springer, 1994 MR 1299725 | Zbl 0943.93001
[16] M. Hirsch, Differential topology, Springer, 1976 MR 448362 | Zbl 0356.57001
[17] S.-Z. Huang, Gradient inequalities with applications to asymptotic behavior and stability of gradient-like systems 126, AMS Mathematical Surveys and Monographs, 2006 MR 2226672 | Zbl 1132.35002
[18] D. Jiang & J. B. Moore, “A gradient flow approach to decentralised output feedback optimal control”, Systems Control Lett. 27 (1996) no. 4, p. 223-231
Article | MR 1389555 | Zbl 0875.93020
[19] K. Kurdyka, “On gradients of functions definable in o-minimal structures”, Ann. Inst. Fourier (Grenoble) 48 (1998) no. 3, p. 769-783
Cedram | MR 1644089 | Zbl 0934.32009
[20] K. Kurdyka, T. Mostowski & A. Parusiński, “Proof of the Gradient Conjecture of R. Thom”, Ann. of Math. 152 (2000), p. 163-792
Article | MR 1815701 | Zbl 1053.37008
[21] K. Kurdyka & A. Parusiński, “$w_f$-stratification of subanalytic functions and the Łojasiewicz inequality”, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994) no. 2, p. 129-133 Zbl 0799.32007
[22] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in B. Malgrange (Paris 1962). Publications du CNRS, ed., Colloques internationaux du CNRS. Les équations aux dérivées partielles, 1963 Zbl 0234.57007
[23] S. Łojasiewicz, Ensembles semi-analytiques, I.H.E.S. Bures-sur-Yvette, 1965
[24] S. Łojasiewicz, Sur les trajectoires du gradient d’une fonction analytique, in Seminari di Geometria, 1982-1983, p. 115-117 MR 771152 | Zbl 0606.58045
[25] S. Łojasiewicz, “Sur la géométrie semi- et sous- analytique”, Ann. Inst. Fourier (Grenoble) 43 (1993) no. 5, p. 1575-1595
Cedram | MR 1275210 | Zbl 0803.32002
[26] Y. C. Lu, Singularity theory and an introduction to catastrophe theory, Springer, 1976 MR 461562 | Zbl 0354.58008
[27] V. Magnani, “A Blow-up theorem for regular hypersurfaces on nilpotent groups”, Manuscripta Math. 110 (2003) no. 1, p. 55-76
Article | MR 1951800 | Zbl 1010.22010
[28] J. H. Manton, U. Helmke & I. M. Y. Mareels, “A dual purpose principal and minor component flow”, Systems Control Lett. 54 (2005) no. 8, p. 759-769
Article | MR 2147235 | Zbl 1129.34319
[29] P. K. Rashevsky, “Any two points of a totally nonholonomic space may be connected by an admissible line”, Uch. Zap. Ped. Inst. im. Liebknechta. Ser. Phys. Math. 2 (1938), p. 83-94
[30] D. Ridout & K. Judd, “Convergence properties of gradient descent noise reduction”, Physica. D 165 (2002) no. 1-2, p. 26-47
Article | MR 1910616 | Zbl 1008.37049
[31] L. Simon, “Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems”, Ann. of Math. (2) 118 (1983) no. 3, p. 525-571
Article | MR 727703 | Zbl 0549.35071
[32] R. S. Strichartz, “Sub-riemannian geometry”, J. Diff. Geom. 24 (1986), p. 221-263
Article | MR 862049 | Zbl 0609.53021
[33] H. J. Sussmann, “Orbits of families of vector fields and integrability of distributions”, Trans. Amer. Math. Soc. 180 (1973), p. 171-188
Article | MR 321133 | Zbl 0274.58002
[34] R. Thom, “Problèmes rencontrés dans mon parcours mathématiques : un bilan”, Publ. Math. IHES 70 (1989), p. 200-214
Numdam | MR 1067383 | Zbl 0709.58001
[35] N. T. Trendafilov & R. A. Lippert, “The multimode Procrustes problem”, Linear Algebra Appl. 349 (2002), p. 245-264
Article | MR 1903736 | Zbl 0999.65051
[36] W. Y. Yan, K. L. Teo & J. B. Moore, “A gradient flow approach to computing LQ optimal output feedback gains”, Optimal Control Appl. Methods 15 (1994) no. 1, p. 67-75
Article | MR 1263418 | Zbl 0815.49025
[37] S. Yoshizawa, U. Helmke & K. Starkow, “Convergence analysis for principal component flows”, Mathematical theory of networks and systems (Perpignan, 2000). Int. J. Appl. Math. Comput. Sci. 11 (2001) no. 1, p. 223-236 MR 1835155 | Zbl pre01599128
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