Field theories in the modern calculus of variations

Nowakowski, Andrzej

doi:10.1090/S0002-9947-1988-0961610-5

Field theories in the modern calculus of variations
HTML articles powered by AMS MathViewer

by Andrzej Nowakowski PDF

Trans. Amer. Math. Soc. 309 (1988), 725-752 Request permission

Abstract:

Two methods of construction of fields of extremals ("geodesic coverings") in the generalized problem of Bolza are given and, as a consequence, sufficient conditions for optimality in a form similar to Weierstrass’ are formulated. The first field theory is an extension of Young’s field theory— "concourse of flights" for our problem; the other describes a nonclassical treatment of field theory which allows one to reject the "self-multiplier restriction".

References

Variationsrechnung und partielle differential Gleichungen erster ordnung

C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262. MR 367131, DOI 10.1090/S0002-9947-1975-0367131-6
Frank H. Clarke, Necessary conditions for a general control problem, Calculus of variations and control theory (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; dedicated to Laurence Chisholm Young on the occasion of his 70th birthday), Publ. Math. Res. Center Univ. Wisconsin, No. 36, Academic Press, New York, 1976, pp. 257–278. MR 0638210
Frank H. Clarke, Extremal arcs and extended Hamiltonian systems, Trans. Amer. Math. Soc. 231 (1977), no. 2, 349–367. MR 442784, DOI 10.1090/S0002-9947-1977-0442784-4
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc. 289 (1985), no. 1, 73–98. MR 779053, DOI 10.1090/S0002-9947-1985-0779053-3
Phillip A. Griffiths, Exterior differential systems and the calculus of variations, Progress in Mathematics, vol. 25, Birkhäuser, Boston, Mass., 1983. MR 684663
Marston Morse, The calculus of variations in the large, American Mathematical Society Colloquium Publications, vol. 18, American Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original. MR 1451874, DOI 10.1090/coll/018
A. Nowakowski, Sufficient conditions for a strong relative minimum in an optimal control problem, J. Optim. Theory Appl. 50 (1986), no. 1, 129–147. MR 851130, DOI 10.1007/BF00938481
R. Tyrrell Rockafellar, Existence theorems for general control problems of Bolza and Lagrange, Advances in Math. 15 (1975), 312–333. MR 365273, DOI 10.1016/0001-8708(75)90140-1
L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. Foreword by Wendell H. Fleming. MR 0259704
Vera Zeidan, Sufficient conditions for the generalized problem of Bolza, Trans. Amer. Math. Soc. 275 (1983), no. 2, 561–586. MR 682718, DOI 10.1090/S0002-9947-1983-0682718-3