The semigroup property of value functions in Lagrange problems

Author:
Peter R. Wolenski

Journal:
Trans. Amer. Math. Soc. **335** (1993), 131-154

MSC:
Primary 49J52; Secondary 49K15, 49L05

DOI:
https://doi.org/10.1090/S0002-9947-1993-1156301-6

MathSciNet review:
1156301

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Abstract | References | Similar Articles | Additional Information

Abstract: The Lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form. The binary operation of inf-composition applied to the value functions of a Lagrange problem equates the principle of optimality with a semigroup property. This paper finds the infinitesimal generator of the semigroup by differentiating at . The type of limit is epigraphical convergence in a uniform sense. Moreover, the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed. The main results provide a new approach to existence and uniqueness questions in Hamilton-Jacobi theory. When is in addition finite-valued, the results are given in terms of pointwise convergence.

**[1]**R. Bellman,*Dynamic programming*, Princeton Univ. Press, Princeton, N.J., 1957. MR**0090477 (19:820d)****[2]**L. D. Berkowitz,*Optimal feedback controls*, SIAM J. Control Optim.**27**(1989), 991-1006. MR**1009334 (90h:49013)****[3]**L. Cesari,*Optimization--Theory and Applications*, Springer-Verlag, New York, 1983. MR**688142 (85c:49001)****[4]**F. H. Clarke,*Extremal arcs and extended Hamiltonian systems*, Trans. Amer. Math. Soc.**231**(1977), 349-367. MR**0442784 (56:1163)****[5]**-,*Optimization and nonsmooth analysis*, Wiley Interscience, New York, 1983. MR**709590 (85m:49002)****[6]**F. H. Clarke and R. B. Vinter,*Local optimality conditions and Lipschitzian solutions to the Hamilton-Jacobi equation*, SIAM J. Control Optim.**21**(1983), 856-870. MR**719517 (85c:49012)****[7]**-,*Regularity properties of solutions to the basic problem in the calculus of variations*, Trans. Amer. Math. Soc.**289**(1985), 73-98. MR**779053 (86h:49020)****[8]**-,*Existence and regularity in the small in the calculus of variations*, J. Differential Equations**59**(1985), 336-354. MR**807852 (87a:49014)****[9]**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**732102 (86a:35031)****[10]**M. G. Crandall, H. Ishii and P. L. Lions,*Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisted*, J. Math. Soc. Japan**39**(1987), 581-595. MR**905626 (88k:35038)****[11]**M. G. Crandall and P. L. Lions,*Viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc.**277**(1983), 1-42. MR**690039 (85g:35029)****[12]**-,*On existence and uniqueness of solutions of Hamilton-Jacobi equations*, Nonlinear Analysis, Theory, Methods and Applications**10**(1986), 353-370. MR**836671 (87f:35052)****[13]**A. F. Filippov,*Classical solutions of differential equations with multivalued right-hand side*, SIAM J. Control Optim.**5**(1967), 609-621. MR**0220995 (36:4047)****[14]**H. Frankowska,*Optimal trajectories associated to a solution of the contingent Hamilton-Jacobi equation*, Appl. Math. Opt.**19**(1989), 291-311. MR**974188 (90b:49043)****[15]**R. L. Gonzales,*Sur l'existence d'une solution maximale de l'équation de Hamilton-Jacobi*, C. R. Acad. Sci. Paris**282**(1976), 1287-1290.**[16]**H. Ishii,*Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations*, Indiana Univ. Math. J.**33**(1984), 721-748. MR**756156 (85h:35057)****[17]**P. L. Lions,*Generalized solutions of Hamilton-Jacobi equations*, Pitman, London, 1982. MR**667669 (84a:49038)****[18]**P. L. Lions and P. E. Souganidis,*Differential games, optimal control, and directional derivatives of viscosity solutions of Bellman's and Isaac's equations*, SIAM J. Control Optim.**23**(1985), 566-583. MR**791888 (87c:49038)****[19]**A. I. Panasiuk and V. I. Panasiuk,*On one equation resulting from a differential inclusion*, Mat. Zametki**27**(1980), 429-445. (Russian)**[20]**L. S. Pontryagin et al.,*The mathematical theory of optimal processes*, Interscience, New York, 1962.**[21]**R. T. Rockafellar,*Convex analysis*, Princeton Univ. Press, Princeton, N. J., 1970.**[22]**-,*Optimal arcs and the minimum value function in problems of Lagrange*, Trans. Amer. Math. Soc.**180**(1973), 53-83. MR**0320852 (47:9385)****[23]**-,*Semigroups of convex bifunctions generated by Lagrange problems in the calculus of variations*, Math. Scand.**36**(1975), 137-158. MR**0385567 (52:6428)****[24]**-,*Existence theorems for general control problems of Bolza and Lagrange*, Adv. in Math.**15**(1975), 312-333. MR**0365273 (51:1526)****[25]**-,*Integral functionals, normal integrands, and measurable selections*, Nonlinear Operators and the Calculus of Variations, (L. Waelbroeck, Ed.), Lecture Notes in Math., vol. 543, Springer-Verlag, 1976, pp. 157-207. MR**0512209 (58:23598)****[26]**E. Roxin,*On the generalized dynamical systems defined by contingent equation*, J. Differential Equations**1**(1965), 188-205. MR**0201756 (34:1638)****[27]**R. B. Vinter and R. M. Lewis,*A necessary and sufficent condition for optimality of dynamic programming type, making no a priori assumptions on the controls*, SIAM J. Control Optim.**16**(1978), 571-583. MR**0493636 (58:12622)****[28]**R. B. Vinter and P. R. Wolenski,*Hamilton-Jacobi theory for optimal control problems with data measurable in time*, SIAM J. Control Optim. (to appear). MR**1075209 (91k:49044)****[29]**R. J. B. Wets,*Convergence of convex functions, variational inequalities, and convex optimization problems*, Variational Inequalities and Complementarity Problems, (R. Cottle, F. Giannessi, and J. L. Lions, Eds.), Wiley, 1980, pp. 375-403. MR**578760 (83a:90140)****[30]**P. R. Wolenski,*Semigroups of multifunctions and properties of the value function*, Thesis, Univ. of Washington, 1988.**[31]**-,*The exponential formula for the reachable set of a Lipschitz differential inclusion*, SIAM J. Control Optim. (to appear). MR**1064723 (91j:93005)****[32]**-,*A uniqueness theorem for differential inclusions*, J. Differential Equations (to appear). MR**1042664 (91c:49009)****[33]**V. Zeidan,*Sufficient conditions for the generalized problem of Bolza*, Trans. Amer. Math. Soc.**275**(1983), 561-586. MR**682718 (84c:49033)****[34]**-,*A modified Hamilton-Jacobi approach in the generalized problem of Bolza*, Appl. Math. Optim.**11**(1984), 97-109. MR**743921 (85m:49056)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1156301-6

Keywords:
Lagrange problems,
principle of optimality,
epigraphical convergence,
Hamilton-Jacobi theory

Article copyright:
© Copyright 1993
American Mathematical Society