Generalized gradients and applications

Clarke, Frank H.

doi:10.1090/S0002-9947-1975-0367131-6

Generalized gradients and applications
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by Frank H. Clarke PDF

Trans. Amer. Math. Soc. 205 (1975), 247-262 Request permission

Abstract:

A theory of generalized gradients for a general class of functions is developed, as well as a corresponding theory of normals to arbitrary closed sets. It is shown how these concepts subsume the usual gradients and normals of smooth functions and manifolds, and the subdifferentials and normals of convex analysis. A theorem is proved concerning the differentiability properties of a function of the form $\max \{ g(x,u):u \in U\}$. This result unifies and extends some theorems of Danskin and others. The results are then applied to obtain a characterization of flow-invariant sets which yields theorems of Bony and Brezis as corollaries.

References

Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 277–304 xii (French, with English summary). MR 262881
Haïm Brezis, On a characterization of flow-invariant sets, Comm. Pure Appl. Math. 23 (1970), 261–263. MR 257511, DOI 10.1002/cpa.3160230211

Necessary conditions for nonsmooth problems in optimal control and the calculus of variations

Necessary conditions for nonsmooth variational problems

V. F. Dem′janov and V. N. Malozemov, The theory of nonlinear minimax problems, Uspehi Mat. Nauk 26 (1971), no. 3(159), 53–104 (Russian). MR 0297378
John M. Danskin, The theory of max-min and its application to weapons allocation problems, Econometrics and Operations Research, V, Springer-Verlag New York, Inc., New York, 1967. MR 0228260
A. F. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control 5 (1967), 609–621. MR 0220995
William Hogan, Directional derivatives for extremal-value functions with applications to the completely convex case, Operations Res. 21 (1973), 188–209. Mathematical programming and its applications. MR 351463, DOI 10.1287/opre.21.1.188
B. N. Pshenichnyi, Necessary conditions for an extremum, Pure and Applied Mathematics, vol. 4, Marcel Dekker, Inc., New York, 1971. Translated from the Russian by Karol Makowski; Translation edited by Lucien W. Neustadt. MR 0276845
R. M. Redheffer, The theorems of Bony and Brezis on flow-invariant sets, Amer. Math. Monthly 79 (1972), 740–747. MR 303024, DOI 10.2307/2316263
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
R. T. Rockafellar, Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl. 32 (1970), 174–222. MR 266020, DOI 10.1016/0022-247X(70)90324-0
R. T. Rockafellar, Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc. 159 (1971), 1–40. MR 282283, DOI 10.1090/S0002-9947-1971-0282283-0
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095