메뉴 건너뛰기




Volumn 173, Issue 816, 2005, Pages

Necessary conditions in dynamic optimization

Author keywords

Calculus of variations; Differential inclusion; Maximum principle; Nonsmooth analysis; Optimal control

Indexed keywords


EID: 12744254041     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0816     Document Type: Article
Times cited : (87)

References (75)
  • 1
    • 0001135181 scopus 로고
    • One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation
    • J. M. Ball and V. J. Mizel. One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal., 90:325-388, 1985.
    • (1985) Arch. Ration. Mech. Anal. , vol.90 , pp. 325-388
    • Ball, J.M.1    Mizel, V.J.2
  • 3
    • 0035283859 scopus 로고    scopus 로고
    • Dualization of the Euler and Hamiltonian inclusions
    • D. Bessis, Y. S. Ledyaev, and R. B. Vinter. Dualization of the Euler and Hamiltonian inclusions. Nonlinear Anal., 43:861-882, 2001.
    • (2001) Nonlinear Anal. , vol.43 , pp. 861-882
    • Bessis, D.1    Ledyaev, Y.S.2    Vinter, R.B.3
  • 4
    • 5244283395 scopus 로고
    • Maximum principle for differential inclusions
    • In Russian
    • V. I. Blagodatskih. Maximum principle for differential inclusions. Trudy Mat. Inst. Steklov, 166:23-43, 1984. In Russian; English translation in Proc. Steklov Inst. Math. 166 (1986).
    • (1984) Trudy Mat. Inst. Steklov , vol.166 , pp. 23-43
    • Blagodatskih, V.I.1
  • 5
    • 12744265437 scopus 로고
    • V. I. Blagodatskih. Maximum principle for differential inclusions. Trudy Mat. Inst. Steklov, 166:23-43, 1984. In Russian; English translation in Proc. Steklov Inst. Math. 166 (1986).
    • (1986) Proc. Steklov Inst. Math. , pp. 166
  • 7
    • 12744266723 scopus 로고
    • Local section method in the theory of optimal processes
    • In Russian
    • V. G. Boltianski. Local section method in the theory of optimal processes. Differentsial'nye Uravneniya, 4:2166-2183, 1968. In Russian; English translation in Differential Equations 4 (1968).
    • (1968) Differentsial'nye Uravneniya , vol.4 , pp. 2166-2183
    • Boltianski, V.G.1
  • 8
    • 12744251767 scopus 로고
    • V. G. Boltianski. Local section method in the theory of optimal processes. Differentsial'nye Uravneniya, 4:2166-2183, 1968. In Russian; English translation in Differential Equations 4 (1968).
    • (1968) Differential Equations , vol.4
  • 9
    • 84966213239 scopus 로고
    • A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions
    • J. M. Borwein and D. Preiss. A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Amer. Math. Soc., 303:517-527, 1987.
    • (1987) Trans. Amer. Math. Soc. , vol.303 , pp. 517-527
    • Borwein, J.M.1    Preiss, D.2
  • 10
    • 0033285086 scopus 로고    scopus 로고
    • A survey of subdifferential calculus with applications
    • J. M. Borwein and Q. J. Zhu. A survey of subdifferential calculus with applications. Nonlinear Anal., 38:687-773, 1999.
    • (1999) Nonlinear Anal. , vol.38 , pp. 687-773
    • Borwein, J.M.1    Zhu, Q.J.2
  • 11
    • 12744258277 scopus 로고    scopus 로고
    • PhD thesis, Université Claude Bernard Lyon I, (Directeur: Francis Clarke).
    • P. Bousquet. Free-time problems in dynamic optimization. PhD thesis, Université Claude Bernard Lyon I, 2003. (Directeur: Francis Clarke).
    • (2003) Free-time Problems in Dynamic Optimization
    • Bousquet, P.1
  • 15
    • 0016557981 scopus 로고
    • Admissible relaxation in variational and control problems
    • F. H. Clarke. Admissible relaxation in variational and control problems. J. Math. Anal. Appl., 51:557-576, 1975.
    • (1975) J. Math. Anal. Appl. , vol.51 , pp. 557-576
    • Clarke, F.H.1
  • 16
    • 0002230594 scopus 로고
    • The Euler-Lagrange differential inclusion
    • F. H. Clarke. The Euler-Lagrange differential inclusion. J. Differential Equations, 19:80-90, 1975.
    • (1975) J. Differential Equations , vol.19 , pp. 80-90
    • Clarke, F.H.1
  • 17
    • 84968484621 scopus 로고
    • Maximum principles without differentiability
    • F. H. Clarke. Maximum principles without differentiability. Bulletin Amer. Math. Soc., 81:219-222, 1975.
    • (1975) Bulletin Amer. Math. Soc. , vol.81 , pp. 219-222
    • Clarke, F.H.1
  • 18
    • 0141812155 scopus 로고
    • The generalized problem of Bolza
    • F. H. Clarke. The generalized problem of Bolza. SIAM J. Control Optim., 14:682-699, 1976.
    • (1976) SIAM J. Control Optim. , vol.14 , pp. 682-699
    • Clarke, F.H.1
  • 19
    • 0017016966 scopus 로고
    • The maximum principle under minimal hypotheses
    • F. H. Clarke. The maximum principle under minimal hypotheses. SIAM J. Control Optim., 14:1078-1091, 1976.
    • (1976) SIAM J. Control Optim. , vol.14 , pp. 1078-1091
    • Clarke, F.H.1
  • 20
    • 0016973122 scopus 로고
    • Optimal solutions to differential inclusions
    • F. H. Clarke. Optimal solutions to differential inclusions. J. Optim. Th. Appl., 19:469-478, 1976.
    • (1976) J. Optim. Th. Appl. , vol.19 , pp. 469-478
    • Clarke, F.H.1
  • 21
    • 0242627486 scopus 로고
    • Extremal arcs and extended Hamiltonian systems
    • F. H. Clarke. Extremal arcs and extended Hamiltonian systems. Trans. Amer. Math. Soc., 231:349-367, 1977. Russian translation: Differentsialnye Uravnenia 13 (1977) 427-442.
    • (1977) Trans. Amer. Math. Soc. , vol.231 , pp. 349-367
    • Clarke, F.H.1
  • 22
    • 0242627486 scopus 로고
    • Russian translation
    • F. H. Clarke. Extremal arcs and extended Hamiltonian systems. Trans. Amer. Math. Soc., 231:349-367, 1977. Russian translation: Differentsialnye Uravnenia 13 (1977) 427-442.
    • (1977) Differentsialnye Uravnenia , vol.13 , pp. 427-442
  • 23
    • 0003859933 scopus 로고
    • Wiley-Interscience, New York
    • F. H. Clarke. Optimization and Nonsmooth Analysis. Wiley-Interscience, New York, 1983. Republished as vol.5 of Classics in Applied Mathematics, SIAM, 1990.
    • (1983) Optimization and Nonsmooth Analysis
    • Clarke, F.H.1
  • 24
    • 0005658675 scopus 로고
    • SIAM
    • F. H. Clarke. Optimization and Nonsmooth Analysis. Wiley-Interscience, New York, 1983. Republished as vol.5 of Classics in Applied Mathematics, SIAM, 1990.
    • (1990) Classics in Applied Mathematics , vol.5
  • 25
    • 84967773289 scopus 로고
    • Hamiltonian analysis of the generalized problem of Bolza
    • F. H. Clarke. Hamiltonian analysis of the generalized problem of Bolza. Trans. Amer. Math. Soc., 301:385-400, 1987.
    • (1987) Trans. Amer. Math. Soc. , vol.301 , pp. 385-400
    • Clarke, F.H.1
  • 26
    • 12744251768 scopus 로고
    • Regularity, existence and necessary conditions for the basic problem in the calculus of variations
    • L. Cesari, editor, London, Longman.
    • F. H. Clarke. Regularity, existence and necessary conditions for the basic problem in the calculus of variations. In L. Cesari, editor, Contributions to the modern calculus of variations, London, 1987. Longman.
    • (1987) Contributions to the Modern Calculus of Variations
    • Clarke, F.H.1
  • 27
    • 12744264143 scopus 로고
    • S.I.A.M., Philadelphia, Regional conference Series in Applied Mathematics
    • F. H. Clarke. Methods of Dynamic and Nonsmooth Optimization. S.I.A.M., Philadelphia, 1989. Regional conference Series in Applied Mathematics vol. 57.
    • (1989) Methods of Dynamic and Nonsmooth Optimization , vol.57
    • Clarke, F.H.1
  • 28
    • 0002250161 scopus 로고
    • A decoupling principle in the calculus of variations
    • F. H. Clarke. A decoupling principle in the calculus of variations. J. Math. Anal. Appl., 172:92-105, 1993.
    • (1993) J. Math. Anal. Appl. , vol.172 , pp. 92-105
    • Clarke, F.H.1
  • 29
    • 0006484428 scopus 로고
    • An indirect method in the calculus of variations
    • F. H. Clarke. An indirect method in the calculus of variations. Trans. Amer. Math. Soc., 336:655-673, 1993.
    • (1993) Trans. Amer. Math. Soc. , vol.336 , pp. 655-673
    • Clarke, F.H.1
  • 30
    • 0012751750 scopus 로고    scopus 로고
    • Nonsmooth analysis in control theory: A survey
    • F. H. Clarke. Nonsmooth analysis in control theory: a survey. European J. Control, 7:63-78, 2001.
    • (2001) European J. Control , vol.7 , pp. 63-78
    • Clarke, F.H.1
  • 31
    • 12744270504 scopus 로고    scopus 로고
    • Necessary conditions in optimal control: A new approach
    • F. H. Clarke. Necessary conditions in optimal control: a new approach. Math. Program., Ser.B, 97:71-89, 2003.
    • (2003) Math. Program., Ser.B , vol.97 , pp. 71-89
    • Clarke, F.H.1
  • 32
    • 84968476212 scopus 로고
    • Mean value inequalities in Hilbert space
    • F. H. Clarke and Yu. S. Ledyaev. Mean value inequalities in Hilbert space. Trans. Amer. Math. Soc., 344:307-324, 1994.
    • (1994) Trans. Amer. Math. Soc. , vol.344 , pp. 307-324
    • Clarke, F.H.1    Ledyaev, Y.S.2
  • 34
    • 84966256938 scopus 로고
    • Regularity properties of solutions to the basic problem in the calculus of variations
    • F. H. Clarke and R. B. Vinter. Regularity properties of solutions to the basic problem in the calculus of variations. Trans. Amer. Math. Soc., 289:73-98, 1985.
    • (1985) Trans. Amer. Math. Soc. , vol.289 , pp. 73-98
    • Clarke, F.H.1    Vinter, R.B.2
  • 35
    • 12744256969 scopus 로고
    • Maximum principle for differential inclusions (necessity)
    • R. P. Fedorenko. Maximum principle for differential inclusions (necessity). Zhurnal Vyc. Matematiki i Matem. Fiziki, 11:885-893, 1971.
    • (1971) Zhurnal Vyc. Matematiki I Matem. Fiziki , vol.11 , pp. 885-893
    • Fedorenko, R.P.1
  • 36
    • 0023090297 scopus 로고
    • The maximum principle for an optimal solution to a differential inclusion with endpoint constraints
    • H. Frankowska. The maximum principle for an optimal solution to a differential inclusion with endpoint constraints. SIAM J. Control Optim., 25:145-157, 1987.
    • (1987) SIAM J. Control Optim. , vol.25 , pp. 145-157
    • Frankowska, H.1
  • 37
    • 0036033314 scopus 로고    scopus 로고
    • Cosmically Lipschitz set-valued mappings
    • G. N. Galbraith. Cosmically Lipschitz set-valued mappings. Set-Valued Analysis, 10:331-360, 2002.
    • (2002) Set-Valued Analysis , vol.10 , pp. 331-360
    • Galbraith, G.N.1
  • 40
    • 84933155131 scopus 로고
    • Extremal properties of biconvex contingent equations
    • Academic Press, New York
    • H. Halkin. Extremal properties of biconvex contingent equations. In Ordinary Differential Equations (NRL-MRC Conf.). Academic Press, New York, 1972.
    • (1972) Ordinary Differential Equations (NRL-MRC Conf.)
    • Halkin, H.1
  • 42
    • 21744456348 scopus 로고    scopus 로고
    • Euler-Lagrange and Hamiltonian formalisms in dynamic optimization
    • A. D. Ioffe. Euler-Lagrange and Hamiltonian formalisms in dynamic optimization. Trans. Amer. Math. Soc., 349:2871-2900, 1997.
    • (1997) Trans. Amer. Math. Soc. , vol.349 , pp. 2871-2900
    • Ioffe, A.D.1
  • 43
    • 0001812895 scopus 로고    scopus 로고
    • The Euler and Weierstrass conditions for nonsmooth variational problems
    • A.D. Ioffe and R. T. Rockafellar. The Euler and Weierstrass conditions for nonsmooth variational problems. Calc. Var. Partial Differential Eq., 4:59-87, 1996.
    • (1996) Calc. Var. Partial Differential Eq. , vol.4 , pp. 59-87
    • Ioffe, A.D.1    Rockafellar, R.T.2
  • 45
    • 0022012587 scopus 로고
    • A maximum principle for generalized control systems
    • B. Kaskosz and S. Lojasiewicz. A maximum principle for generalized control systems. Nonlinear Anal., 9:109-130, 1985.
    • (1985) Nonlinear Anal. , vol.9 , pp. 109-130
    • Kaskosz, B.1    Lojasiewicz, S.2
  • 46
    • 0026925456 scopus 로고
    • Lagrange-type extremal trajectories in differential inclusions
    • B. Kaskosz and S. Lojasiewicz. Lagrange-type extremal trajectories in differential inclusions. Syst. Control Letters, 19:241-247, 1992.
    • (1992) Syst. Control Letters , vol.19 , pp. 241-247
    • Kaskosz, B.1    Lojasiewicz, S.2
  • 50
    • 0028396661 scopus 로고
    • Optimal control of unbounded differential inclusions
    • P. D. Loewen and R. T. Rockafellar. Optimal control of unbounded differential inclusions. SIAM J. Control Optim., 32:442-470, 1994.
    • (1994) SIAM J. Control Optim. , vol.32 , pp. 442-470
    • Loewen, P.D.1    Rockafellar, R.T.2
  • 51
    • 0030243608 scopus 로고    scopus 로고
    • New necessary conditions for the generalized problem of Bolza
    • P. D. Loewen and R. T. Rockafellar. New necessary conditions for the generalized problem of Bolza. SIAM J. Control Optim., 34:1496-1511, 1996.
    • (1996) SIAM J. Control Optim. , vol.34 , pp. 1496-1511
    • Loewen, P.D.1    Rockafellar, R.T.2
  • 52
    • 0007185642 scopus 로고
    • On multipliers for Lagrange problems
    • E. McShane. On multipliers for Lagrange problems. American J. Math., 61:809-819, 1939.
    • (1939) American J. Math. , vol.61 , pp. 809-819
    • McShane, E.1
  • 54
    • 85052489384 scopus 로고    scopus 로고
    • Strengthening the conditions of Clarke and Smirnov for convex-valued differential inclusions
    • A. A. Milyutin. Strengthening the conditions of Clarke and Smirnov for convex-valued differential inclusions. Matematicheskii Sbornik, 194:87-116, 2003.
    • (2003) Matematicheskii Sbornik , vol.194 , pp. 87-116
    • Milyutin, A.A.1
  • 56
    • 84966250815 scopus 로고
    • Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions
    • B. S. Mordukhovich. Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc., 340:1-35, 1993.
    • (1993) Trans. Amer. Math. Soc. , vol.340 , pp. 1-35
    • Mordukhovich, B.S.1
  • 57
    • 0029307052 scopus 로고
    • Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions
    • B. S. Mordukhovich. Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim., 33:882-915, 1995.
    • (1995) SIAM J. Control Optim. , vol.33 , pp. 882-915
    • Mordukhovich, B.S.1
  • 60
    • 84972555015 scopus 로고
    • Generalized Hamiltonian equations for convex problems of Lagrange
    • R. T. Rockafellar. Generalized Hamiltonian equations for convex problems of Lagrange. Pacific J. Math., 33:411-428, 1970.
    • (1970) Pacific J. Math. , vol.33 , pp. 411-428
    • Rockafellar, R.T.1
  • 61
    • 0000579576 scopus 로고
    • Existence theorems for general control problems of Bolza and Lagrange
    • R. T. Rockafellar. Existence theorems for general control problems of Bolza and Lagrange. Advances in Math., 15:312-333, 1975.
    • (1975) Advances in Math. , vol.15 , pp. 312-333
    • Rockafellar, R.T.1
  • 62
    • 0021938097 scopus 로고
    • Extensions of subgradient calculus with applications to optimization
    • R. T. Rockafellar. Extensions of subgradient calculus with applications to optimization. Nonlinear Anal., 9:665-698, 1985.
    • (1985) Nonlinear Anal. , vol.9 , pp. 665-698
    • Rockafellar, R.T.1
  • 63
    • 0030191882 scopus 로고    scopus 로고
    • Equivalent subgradient versions of Hamiltonian and Euler-Lagrange equations in variational analysis
    • R. T. Rockafellar. Equivalent subgradient versions of Hamiltonian and Euler-Lagrange equations in variational analysis. SIAM J. Control Optim., 34:1300-1314, 1996.
    • (1996) SIAM J. Control Optim. , vol.34 , pp. 1300-1314
    • Rockafellar, R.T.1
  • 65
    • 0025927009 scopus 로고
    • Discrete approximations and optimal solutions to differential inclusions
    • In Russian
    • G. V. Smirnov. Discrete approximations and optimal solutions to differential inclusions. Kibernetika, 10:76-79, 1991. In Russian; English translation in Cybernetics, 27 (1991), pp. 101-107.
    • (1991) Kibernetika , vol.10 , pp. 76-79
    • Smirnov, G.V.1
  • 66
    • 0043028111 scopus 로고
    • G. V. Smirnov. Discrete approximations and optimal solutions to differential inclusions. Kibernetika, 10:76-79, 1991. In Russian; English translation in Cybernetics, 27 (1991), pp. 101-107.
    • (1991) Cybernetics , vol.27 , pp. 101-107
  • 67
    • 0002671270 scopus 로고    scopus 로고
    • Geometry and optimal control
    • J. Baillieul and J. C. Willems, editors, New York, Springer-Verlag.
    • H. J. Sussmann. Geometry and optimal control. In J. Baillieul and J. C. Willems, editors, Mathematical Control Theory, pages 140-194, New York, 1999. Springer-Verlag.
    • (1999) Mathematical Control Theory , pp. 140-194
    • Sussmann, H.J.1
  • 70
    • 21944450668 scopus 로고    scopus 로고
    • The extended Euler-Lagrange condition for nonconvex variational problems with state constraints
    • R. B. Vinter and H. Zheng. The extended Euler-Lagrange condition for nonconvex variational problems with state constraints. Trans. Amer. Math. Soc., 350:1181-1204, 1998.
    • (1998) Trans. Amer. Math. Soc. , vol.350 , pp. 1181-1204
    • Vinter, R.B.1    Zheng, H.2
  • 73
    • 0001312592 scopus 로고
    • Sufficient conditions for the generalized problem of Bolza
    • V. Zeidan. Sufficient conditions for the generalized problem of Bolza. Trans. Amer. Math. Soc., 275:561-586, 1983.
    • (1983) Trans. Amer. Math. Soc. , vol.275 , pp. 561-586
    • Zeidan, V.1
  • 75
    • 0029715234 scopus 로고    scopus 로고
    • Necessary optimality conditions for nonconvex differential inclusions with endpoint constraints
    • Q. J. Zhu. Necessary optimality conditions for nonconvex differential inclusions with endpoint constraints. J. Differential Eq., 124:186-204, 1996.
    • (1996) J. Differential Eq. , vol.124 , pp. 186-204
    • Zhu, Q.J.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.