Sufficient conditions for the generalized problem of Bolza

Zeidan, Vera

doi:10.1090/S0002-9947-1983-0682718-3

Sufficient conditions for the generalized problem of Bolza
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Trans. Amer. Math. Soc. 275 (1983), 561-586 Request permission

Abstract:

This paper presents sufficient conditions for strong local optimality in the generalized problem of Bolza. These conditions represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data. Also, this paper brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations, showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original Hamiltonian to a locally concave-convex Hamiltonian.

References

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, Admissible relaxation in variational and control problems, J. Math. Anal. Appl. 51 (1975), no. 3, 557–576. MR 407676, DOI 10.1016/0022-247X(75)90107-9

L’Hamiltonien en optimisation

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, The generalized problem of Bolza, SIAM J. Control Optim. 14 (1976), no. 4, 682–699. MR 412926, DOI 10.1137/0314044
Frank H. Clarke, Generalized gradients of Lipschitz functionals, Adv. in Math. 40 (1981), no. 1, 52–67. MR 616160, DOI 10.1016/0001-8708(81)90032-3
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
I. M. Gelfand and S. V. Fomin, Calculus of variations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. Revised English edition translated and edited by Richard A. Silverman. MR 0160139
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Magnus R. Hestenes, Calculus of variations and optimal control theory, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0203540
D. Q. Mayne, Sufficient conditions for a control to be a strong minimum, J. Optim. Theory Appl. 21 (1977), no. 3, 339–351. MR 458283, DOI 10.1007/BF00933535

A Hamilton-Jacobi approach to the differential inclusion problem

Stephen M. Robinson, Generalized equations and their solutions. II. Applications to nonlinear programming, Math. Programming Stud. 19 (1982), 200–221. Optimality and stability in mathematical programming. MR 669732, DOI 10.1007/bfb0120989

Convex analysis

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. Tyrrell Rockafellar, Generalized Hamiltonian equations for convex problems of Lagrange, Pacific J. Math. 33 (1970), 411–427. MR 276853
R. Tyrrell Rockafellar, Optimal arcs and the minimum value function in problems of Lagrange, Trans. Amer. Math. Soc. 180 (1973), 53–83. MR 320852, DOI 10.1090/S0002-9947-1973-0320852-1
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
Atle Seierstad and Knut Sydsaeter, Sufficient conditions in optimal control theory, Internat. Econom. Rev. 18 (1977), no. 2, 367–391. MR 454795, DOI 10.2307/2525753