Extremal arcs and extended Hamiltonian systems
Author:
Frank H. Clarke
Journal:
Trans. Amer. Math. Soc. 231 (1977), 349367
MSC:
Primary 49A05
MathSciNet review:
0442784
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Abstract: A general variational problem is considered; it involves the minimization of an integrand L of a very general nature. The Lagrangian L is allowed to assume the value , and need satisfy no differentiability or convexity conditions. A Hamiltonian corresponding to the problem is defined via the conjugate function of convex analysis, and it is shown how one obtains necessary conditions in the form of an extended Hamiltonian system. This system is expressed in terms of certain ``generalized gradients'' previously developed by the author. A further result is given which has the feature that the principal hypotheses required, as well as the ensuing conclusions, are entirely in terms of H. This allows the treatment of classes of problems in which H is more amenable to direct analysis than L. The approach also sheds light on the relation between existence theory and the theory of necessary conditions, since the results may easily be compared with R. T. Rockafellar's recent work on existence theory, in which H also plays a central role. As an example of its application the main result is specialized to a differential inclusion problem. A specific example of its use is also given, an unorthodox optimal control problem with a discontinuous cost functional.
 [1]
Henri
Berliocchi and JeanMichel
Lasry, Principe de Pontryagin pour des systèmes régis
par une équation différentielle multivoque, C. R. Acad.
Sci. Paris Sér. AB 277 (1973), A1103–A1105
(French). MR
0353094 (50 #5580)
 [2]
V.
G. Boltjanskiĭ, On the proof of the maximum principle for
objects with continuous time, Differencial′nye Uravnenija
9 (1973), 1747–1753, 1923 (Russian). MR 0383189
(52 #4070)
 [3]
Frank
H. Clarke, Generalized gradients and
applications, Trans. Amer. Math. Soc. 205 (1975), 247–262.
MR
0367131 (51 #3373), http://dx.doi.org/10.1090/S00029947197503671316
 [4]
Frank
H. Clarke, Admissible relaxation in variational and control
problems, J. Math. Anal. Appl. 51 (1975), no. 3,
557–576. MR 0407676
(53 #11448)
 [5]
Frank
H. Clarke, The generalized problem of Bolza, SIAM J. Control
Optimization 14 (1976), no. 4, 682–699. MR 0412926
(54 #1047)
 [6]
R.
P. Fedorenko, The maximum principle for differential
inclusions, Ž. Vyčisl. Mat. i Mat. Fiz.
10 (1970), 1385–1393 (Russian). MR 0284903
(44 #2127)
 [7]
Hubert
Halkin, Extremal properties of biconvex contingent equations,
Ordinary differential equations (Proc. NRLMRC Conf., Math. Res. Center,
Naval Res. Lab., Washington, D. C., 1971) Academic Press, New York, 1972,
pp. 109–119. MR 0440451
(55 #13326)
 [8]
A.
W. Roberts and D.
E. Varberg, Another proof that convex functions are locally
Lipschitz, Amer. Math. Monthly 81 (1974),
1014–1016. MR 0352371
(50 #4858)
 [9]
R.
Tyrrell Rockafellar, Convex analysis, Princeton Mathematical
Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
(43 #445)
 [10]
R.
T. Rockafellar, Conjugate convex functions in optimal control and
the calculus of variations, J. Math. Anal. Appl. 32
(1970), 174–222. MR 0266020
(42 #929)
 [11]
R.
T. Rockafellar, Existence and duality theorems for
convex problems of Bolza, Trans. Amer. Math.
Soc. 159 (1971),
1–40. MR
0282283 (43 #7995), http://dx.doi.org/10.1090/S00029947197102822830
 [12]
R.
Tyrrell Rockafellar, Existence theorems for general control
problems of Bolza and Lagrange, Advances in Math. 15
(1975), 312–333. MR 0365273
(51 #1526)
 [13]
Michel
Valadier, Existence globale pour les équations
différentielles multivoques, C. R. Acad. Sci. Paris Sér.
AB 272 (1971), A474–A477 (French). MR 0279630
(43 #5351)
 [14]
J.
Warga, Necessary conditions without differentiability assumptions
in optimal control, J. Differential Equations 18
(1975), 41–62. MR 0377655
(51 #13824)
 [1]
 H. Berliocchi and J.M. Lasry, Principe de Pontryagin pour des systèmes régis par une équation différentielle multivoque, C. R. Akad. Sci. Paris Sér. AB 277 (1973), Al103A1105. MR 50 #5580. MR 0353094 (50:5580)
 [2]
 V. G. Boltjanskiĭ, The maximum principle for problems of optimal steering, Differencial'nye Uravnenija 9 (1973), 13631370. MR 0383189 (52:4070)
 [3]
 F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247262. MR 0367131 (51:3373)
 [4]
 , Admissible relaxation in variational and control problems, J. Math. Anal. Appl. 51 (1975), 557576. MR 0407676 (53:11448)
 [5]
 , The generalized problem of Bolza, SIAM J. Control Optimization 14 (1976), 682699. MR 0412926 (54:1047)
 [6]
 R. P. Fedorenko, A maximum principle for differential inclusions, Ž. Vyčisl. Mat. i Mat. Fiz. 10 (1970), 13851393 = USSR Comput. Math. and Math. Phys. 10 (1970), 5768. MR 44 #2127. MR 0284903 (44:2127)
 [7]
 H. Halkin, Extremal properties of biconvex contingent equations, Ordinary Differential Equations (NRLMRC Conf.), Academic Press, New York, 1972, pp. 109120. MR 0440451 (55:13326)
 [8]
 A. W. Roberts and D. E. Varberg, Another proof that convex functions are locally Lipschitz, Amer. Math. Monthly 81 (1974), 10141016. MR 50 #4858. MR 0352371 (50:4858)
 [9]
 R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N. J., 1970. MR 43 #445. MR 0274683 (43:445)
 [10]
 , Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl. 32 (1970), 174222. MR 42 #929. MR 0266020 (42:929)
 [11]
 , Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc. 159 (1971), 140. MR 43 #7995. MR 0282283 (43:7995)
 [12]
 , Existence theorems for general control problems of Bolza and Lagrange, Advances in Math. 15 (1975), 312333. MR 0365273 (51:1526)
 [13]
 M. Valadier, Existence globale pour les équations différentielles multivoques, C. R. Acad. Sci. Paris Sér. AB 272 (1971), A474A477. MR 43 #5351. MR 0279630 (43:5351)
 [14]
 J. Warga, Necessary conditions without differentiability assumptions in optimal control, J. Differential Equations 18 (1975), 4162. MR 0377655 (51:13824)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197704427844
PII:
S 00029947(1977)04427844
Keywords:
Hamiltonian,
maximum principle,
nondifferentiable functions,
differential inclusions,
generalized gradients
Article copyright:
© Copyright 1977
American Mathematical Society
