On the stable equilibrium points of gradient systems

Systems and Control Letters

Volumn 55, Issue 7, 2006, Pages 573-577

  Absil, P A ^a,b   Kurdyka, K ^c  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^c UNIVERSITÉ DE SAVOIE   (France)

Author keywords

Cost function; Gradient flow; Local minimum; Lyapunov stability

Indexed keywords

COMPUTATIONAL METHODS; COSTS; FUNCTIONS; GRADIENT METHODS; SYSTEM STABILITY;

COST FUNCTION; GRADIENT FLOW; LOCAL MINIMUM; LYAPUNOV STABILITY;

LYAPUNOV METHODS;

EID: 33748133454     PISSN: 01676911     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sysconle.2006.01.002     Document Type: Article

References (21)

1. Absil P.-A., Mahony R., and Andrews B. Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 6 2 (2005) 531-547
2. Absil P.-A., and Sepulchre R. Continuous dynamical systems that realize discrete optimization on the hypercube. Systems Control Lett. 52 3-4 (2004) 297-304
3. R. Bachmayer, N.E. Leonard, Vehicle networks for gradient descent in a sampled environment, Proceedings of the 41st IEEE Conference on Decision and Control, 2002.
4. M. Baeg, U. Helmke, J.B. Moore, Gradient flow techniques for pose estimation of quadratic surfaces, Proceedings of the World Congress in Computational Methods and Applied Mathematics, 1994.
5. Bierstone E., and Milman P.D. Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. 67 (1988) 5-42
6. D. D'Acunto, K. Kurdyka, Bounds for gradient trajectories of polynomial and definable functions with applications, 2004, J. Differ. Geom., submitted.
7. Helmke U., and Moore J.B. Optimization and Dynamical Systems (1994), Springer, Berlin
8. Hirsch M.W., and Smale S. Differential Equations, Dynamical Systems, and Linear Algebra (1974), Academic Press, New York
9. Jiang D., and Moore J.B. A gradient flow approach to decentralised output feedback optimal control. Systems Control Lett. 27 4 (1996) 223-231
10. Khalil H.K. Nonlinear Systems. second ed. (1996), Prentice-Hall, Englewood Cliffs, NJ
11. Kurdyka K., Mostowski T., and Parusiński A. Proof of the gradient conjecture of R. Thom. Ann. of Math. (2) 152 3 (2000) 763-792
12. f-stratification of subanalytic functions and the Łojasiewicz inequality. C. R. Acad. Sci. Paris Sér. I Math. 318 2 (1994) 129-133
13. Łojasiewicz S. Ensembles semi-analytiques (1965), Inst. Hautes Études Sci., Bures-sur-Yvette
14. S. Łojasiewicz, Sur les trajectoires du gradient d'une fonction analytique, Seminari di Geometria 1982-1983, Dipartimento di Matematica, Istituto di Geometria, Università di Bologna, 1984, pp. 115-117.
15. Manton J.H., Helmke U., and Mareels I.M.Y. A dual purpose principal and minor component flow. Systems Control Lett. 54 8 (2005) 759-769
16. Ridout D., and Judd K. Convergence properties of gradient descent noise reduction. Physica D 165 (2002) 26-47
17. Rumyantsev V.V., and Sosnitskiǐ S.P. On the instability of the equilibrium of holonomic conservative systems. J. Appl. Math. Mech. 57 6 (1994) 1101-1122
18. Trendafilov N.T., and Lippert R.A. The multimode Procrustes problem. Linear Algebra Appl. 349 (2002) 245-264
19. Wiener Z. Instability of a non-isolated equilibrium. Arch. Rational Mech. Anal. 116 4 (1992) 301-305
20. Yan W.Y., Teo K.L., and Moore J.B. A gradient flow approach to computing LQ optimal output feedback gains. Optim. Control Appl. Methods 15 1 (1994) 67-75
21. Yoshizawa S., Helmke U., and Starkov K. Convergence analysis for principal component flows. Internat. J. Appl. Math. Comput. Sci. 11 1 (2001) 223-236