Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Nonsmooth analysis on smooth manifolds

Author(s): Yu. S. Ledyaev; Qiji J. Zhu
Journal: Trans. Amer. Math. Soc. 359 (2007), 3687-3732.
MSC (2000): Primary 93D05, 93D20, 34D20
Posted: February 23, 2007
MathSciNet review: 2302512
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

References:

1.: J. P. Aubin and A. Cellina.
Differential Inclusions.
Springer-Verlag, Berlin, 1984. MR 0755330 (85j:49010)
2.: J. P. Aubin and H. Frankowska,
Set-Valued Analysis. Birkháuser, Boston, 1990. MR 1048347 (91d:49001)
3.: W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Progress in Mathematics, 61, Birkhuser Boston Inc. Boston MA, 1985. MR 0823981 (87h:53050)
4.: M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, With appendix by M. Falcone and P. Soravia, Systems and Control: Foundations and Applications, Birkhauser Boston, Inc. Boston, MA, 1997.MR 1484411 (99e:49001)
5.: E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Commun. In Partial Differential Equations, 15 (1990), 1713-1742.MR 1080619 (91h:35069)
6.: E. N. Barron and R. Jensen, Optimal control and semicontinuous viscosity solutions, Proc. Amer. Math. Soc. 113 (1991), 396-402. MR 1076572 (91m:49027)
7.: M. S. Bazaraa, J. J. Goode amd M. Z. Nashed, On the cones of tangents with applications to mathematical programming, J. Optim. Theo. Appli., 13 (1974), 389-426. MR 0366398 (51:2645)
8.: J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc., 303 (1987), 517-527. MR 0902782 (88k:49013)
9.: J. M. Borwein, J. Read, A. S. Lewis and Q. J. Zhu, Convex spectral functions of compact operators, J. Nonlinear Convex Anal. 1 (2000), 17-35.MR 1751726 (2003a:49045)
10.: J. M. Borwein, A. S. Lewis and Q. J. Zhu, Convex spectral functions of compact operators, Part 2: Lower semicontinuity and rearragement invariance, Proceedings of the Optimization and related topics (Ballarat/Melbourne, 1999), Applied Optim. 47, Kluwer Academic Publ. 2001, pp. 179-196. MR 1893622 (2004c:49075)
11.: J. M. Borwein, J. S. Treiman and Q. J. Zhu, Necessary conditions for constrained optimization problems with semicontinuous and continuous data, Trans. Amer. Math. Soc. 350 (1998), 2409-2429. MR 1433112 (98h:90108)
12.: J. M. Borwein and Q. J. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control and Optimization, 34 (1996), 1568-1591. MR 1404847 (97g:49037)
13.: J. M. Borwein and Q. J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Analysis, TMA 38 (1999), 687-773. MR 1710152 (2000j:49024)
14.: J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, CMS Springer-Verlag Books, Springer-Verlag, New York, 2005. MR 2144010
15.: J. M. Borwein and D. Zhuang, On Fan's minimax theorem, Mathematical Programming, 34 (1996), 232-234. MR 0838482 (87i:90216)
16.: F. Brickell and R. S. Clark,
Differentiable manifolds: an introduction, V.N. Reinhold Co., London, New York, 1970.
17.: R. W. Brockett, Nonlinear control theory and differential geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 1357-1368. MR 0804784 (86k:93068)
18.: R. W. Brockett,
Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems, Linear Algebra Appl. 146 (1991), 79-91. MR 1083465 (92j:90043)
19.: J. V. Burke and M. L. Overton, On the subdifferentiability of functions of a matrix spectrum. II. Subdifferential formulas, Nonsmooth optimization: methods and applications (Erice, 1991), Gordon and Breach, Montreux, 1992, pp. 19-29. MR 1263488 (95d:49025)
20.: J. V. Burke and M. L. Overton, Differential properties of the spectral abscissa and the spectral radius for analytic matrix-valued mappings, Nonlinear Anal. 23 (1994), 467-488. MR 1294349 (95f:49017)
21.: H. Busemann, The Geometry of Geodesics, Academic Press Inc., New York, N.Y., 1955.MR 0075623 (17:779a)
22.: I. Chryssochoos and R. B. Vinter, Optimal control problems on manifolds: a dynamical programming approach, J. Math. Anal. Appl., 287 (2003) 118-140.MR 2010261 (2004j:49045)
23.: F. H. Clarke, Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations, Ph. D. thesis, Univ. of Washington, 1973.
24.: F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262. MR 0367131 (51:3373)
25.: F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983, Russian edition Nauka, Moscow, 1988; Reprinted as Vol. 5 of the series Classics in Applied Mathematics, SIAM, Philadelphia, 1990.MR 1058436 (91e:49001)
26.: F. H. Clarke, Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional conference series in applied mathematics, Vol. 57 SIAM, Philadelphia, 1989.MR 1085948 (91j:49001)
27.: F. H. Clarke, Yu. S. Ledyaev, E. D. Sontag, and A. I. Subbotin,
Asymptotic controllability implies feedback stabilization, I.E.E.E. Trans. Aut. Control, 42 (1999), 1394-1407.MR 1472857 (98g:93003)
28.: F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Qualitative properties of trajectories of control systems: A survey, J. Dynamical and Contr. Sys. 1 (1995), 1-48.MR 1319056 (95k:49002)
29.: F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics Vol. 178, Springer-Verlag, New York, 1998. MR 1488695 (99a:49001)
30.: F. H. Clarke, R. J. Stern and P. R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Can. J. Math., 45 (1993), 1167-1183. MR 1247540 (94j:49018)
31.: M. G. Crandall, L. C. Evans and P. L. Lions Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. MR 0732102 (86a:35031)
32.: M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1-42. MR 0690039 (85g:35029)
33.: K. Deimling.
Multivalued Differential Equations.
de Gruyter, Berlin, 1992. MR 1189795 (94b:34026)
34.: J. M. Danskin.
The theory of max-min, with applications, SIAM J. Appl. Math., 14 (1966), 641-664. MR 0210456 (35:1349)
35.: J. M. Danskin.
The theory of max-min and its application to weapons allocation problems, Econometrics and Operations Research, Vol. V. Springer-Verlag New York, Inc., New York, 1967. MR 0228260 (37:3843)
36.: V. F. Dem'yanov.
On the directional differentiation of the maximin function, Dokl. Akad. Nauk SSSR, 179 (1968), 1032-1033. MR 0227832 (37:3416)
37.: V. F. Dem'yanov.
Minimax: directional differentiability. Izdat. Leningrad. Univ., Leningrad, 1974.MR 0445825 (56:4159)
38.: V. F. Dem'yanov and V. N. Malozemov, Introduction to minimax, Translated from the Russian by D. Louvish. Reprint of the 1974 edition. Dover Publications, Inc., New York, 1990. MR 1088479 (91k:49001)
39.: I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. MR 0346619 (49:11344)
40.: U. Helmke and J. B. Moore, Optimization and dynamical systems, With a foreword by R. Brockett, Springer-Verlag London Ltd., London, 1994.MR 1299725 (95j:49001)
41.: R. Hermann, Differential Geometry and the Calculus of Variations, Vol. 49 in Mathematics in Science and Engineering, ed. R, Bellman, Academic Press, New York, London, 1968. MR 0233313 (38:1635)
42.: A. D. Ioffe, On subdifferentiability spaces, Ann. N.Y. Acad. Sci., 410 (1983), 107-119. MR 0775520 (86g:90124)
43.: A. D. Ioffe, Subdifferentiablility spaces and nonsmooth analysis, Bull. Amer. Math. Soc. 10 (1984), 87-90.MR 0722857 (85c:58014)
44.: A. D. Ioffe, Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175-192. MR 1063554 (91i:46045)
45.: A. D. Ioffe, Variational analysis in local and global nonsmooth analysis, In F. H. Clarke and R. J. Stern, editors, Nonlinear analysis, differential equations and control, Vol. 528 of NATO Sci. Ser. C Math. Phys Sci., pages 447-502. Kluwer Acad. Publ. Dordrecht, 1999. MR 1695012 (2000e:49022)
46.: A. Isidori, Nonlinear Control Systems, Third Edition, Springer-Verlag, London, 1989. MR 1410988 (97g:93003)
47.: Yu. S. Ledyaev and J.S.Treiman, Sub- and supergradients of envelopes, closures and limits, J.Functional Analysis, submitted.
48.: Yu.S.Ledyaev, J.S.Treiman and Q.J.Zhu, Helly's intersection theorem for manifolds of nonpositive curvature, J.Convex Analysis, submitted.
49.: P. D. Loewen, Optimal Control via Nonsmooth Analysis, CRM Lecture Notes Series, Amer. Math. Soc., Summer School on Control, CRM, Université de Montréal, 1992, Amer. Math. Soc., Providence, 1993.MR 1232864 (94h:49003)
50.: A. Lewis, Nonsmooth analysis of eigenvalues, Mathematical Programming, 84 (1999), 1-24. MR 1687292 (2000d:49029)
51.: Y. Matsushima, Differential Manifolds, Marcel Dekker, New York, 1972. MR 0346831 (49:11553)
52.: P. Michel and J. P. Penot, Calcul sous-différential pour des fonctions Lipschitziennes et non-Lipschiziennes, C. R. Acad. Sci. Paris, Ser. I Math. 298 (1985), 269-272. MR 0745320 (85i:49027)
53.: B. S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960-969.MR 0487669 (58:7284)
54.: B. S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl., 22 (1980), 526-530. MR 0592682 (82b:90104)
55.: B. S. Mordukhovich, Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings, Dokl. Akad. Nauk. BSSR, 28 (1984), 976-979. MR 0771737 (86c:49018)
56.: B. S. Mordukhovich, Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988. (Russian; English transl. to appear in Wiley-Interscience.) MR 0945143 (89m:49001)
57.: B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183 (1994), 250-288.MR 1273445 (95i:49029)
58.: B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I and II, Springer-Verlag, Berlin, 2006. MR 2191744; MR 2191745
59.: B. S. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc., 348 (1996), 1235-1280.MR 1333396 (96h:49036)
60.: B. N. Pshenichnyi, Convex programming in a normalized space, Cybernetics, 1 (1966), 46-57.MR 0234732 (38:3048)
61.: B. N. Pshenichnyi,
Necessary conditions for an extremum.
Marcel Dekker Inc., New York, 1971.
Russian original edition: Nauka, Moscow, 1969. MR 0276845 (43:2585)
62.: R. T. Rockafellar, Convex analysis, Princeton University Press, Princeton, N.J., 1970.MR 0274683 (43:445)
63.: R. T. Rockafellar, Clarke's tangent cones and boundaries of closed sets in $\mathbb{R}^n$ , Nonlinear Analysis: TMA, 3 (1979), 145-154. MR 0520481 (80d:49032)
64.: R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, New York, 1997. MR 1491362 (98m:49001)
65.: S. Sternberg,
Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR 0193578 (33:1797)
66.: A. I. Subbotin, Generalized Solutions of First-Order PDEs: The Dynamical Optimization Perspective, Birkhaüser, Boston, 1995.MR 1320507 (96b:49002)
67.: H. J. Sussmann, A strong version of the maximum principle under weak hypotheses, Proceedings of the 33rd IEEE conference on decision and control, Lake Buena Vista, FL, December 1994.
68.: H. J. Sussmann, A strong maximum principle for systems of differential inclusions, Proceedings of the 35th IEEE conference on decision and control, Kobe, Japan, December 1996.
69.: H. J. Sussmann, Transversality conditions and a strong maximum principle for systems of differential inclusions, Proceedings of the 37th IEEE conference on decision and control, Tempa, FL, December 1998.
70.: J. S. Treiman, Generalized gradients, Lipschitz behavior and directional derivatives, Can. J. Math., 37 (1985), 1074-1084.MR 0828835 (87h:90263)
71.: J. S. Treiman, Finite dimensional optimality conditions: B-gradients, J. Optim. Theory Appl. 62 (1989), 139-150.MR 1006610 (90f:49011)
72.: J. S. Treiman, The linear nonconvex generalized gradients and Lagrange multipliers, SIAM J. Optim. 5 (1995), 670-680.MR 1344675 (96i:49032)
73.: J. S. Treiman, Lagrange multipliers for nonconvex generalized gradients with equality, inequality, and set constraints, SIAM J. Control Optim. 37 (1999), 1313-1329. MR 1710060 (2000j:49026)
74.: R. B. Vinter,
Optimal control, Birkhäuser Boston Inc., Boston, MA, 2000. MR 1756410 (2001c:49001)
75.: J. Warga, Derivate containers, inverse functions, and controllability, in Calculus of Variations and Control Theory, D. L. Russell, Ed., Academic Press, New York, 1976. MR 0427561 (55:592)
76.: J. Warga, Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl. 81 (1981), 545-560. MR 0622836 (83f:58007)
77.: D. Zagrodny, Approximate mean value theorem for upper subderivatives, Nonlinear Anal. TMA, 12 (1988), 1413-1428.MR 0972409 (89k:58034)
78.: Q. J. Zhu, Subderivatives and their applications, Proceedings of International Conference on Dynamical Systems and Differential Equations, Edited by W. Chen and S. Hu, Springfield, MO, June, 1996, pp. 379-394. MR 1722485 (2000j:49027)
79.: Q. J. Zhu, The equivalence of several basic theorems for subdifferentials, Set-Valued Analysis, 6 (1998), 171-185.MR 1646482 (2000c:49035)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|