메뉴 건너뛰기




Volumn 100, Issue 6, 2000, Pages 2613-2632

Optimal control of difference, differential, and differential-difference inclusions

(1)  Mordukhovich, B S a  

a NONE

Author keywords

[No Author keywords available]

Indexed keywords


EID: 0001285530     PISSN: 10723374     EISSN: 15738795     Source Type: Journal    
DOI: 10.1007/BF02672708     Document Type: Article
Times cited : (13)

References (78)
  • 1
    • 4143144565 scopus 로고
    • First order approximations for differential inclusions
    • Z. Artstein, "First order approximations for differential inclusions," Set-Valued Anal, 2, 7-17 (1994).
    • (1994) Set-Valued Anal , vol.2 , pp. 7-17
    • Artstein, Z.1
  • 2
    • 0031140636 scopus 로고    scopus 로고
    • Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints
    • A. V. Arutyunov and S. M. Aseev, "Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints," SIAM J. Control Optim., 35, 930-952 (1997).
    • (1997) SIAM J. Control Optim. , vol.35 , pp. 930-952
    • Arutyunov, A.V.1    Aseev, S.M.2
  • 3
    • 0042093958 scopus 로고    scopus 로고
    • A method of smooth approximations in the theory of necessary optimality conditions for differential inclusions
    • S. M. Aseev, "A method of smooth approximations in the theory of necessary optimality conditions for differential inclusions," Izv. Ross. Akad. Nauk, Ser. Mat., 61, 3-26 (1997);
    • (1997) Izv. Ross. Akad. Nauk, Ser. Mat. , vol.61 , pp. 3-26
    • Aseev, S.M.1
  • 4
    • 21744454217 scopus 로고    scopus 로고
    • English transl.
    • English transl.: Izvestiya: Mathematics, 61, 235-258 (1997).
    • (1997) Izvestiya: Mathematics , vol.61 , pp. 235-258
  • 6
    • 0016114904 scopus 로고
    • Application of abstract variational theory to hereditary systems - A survey
    • H. T. Banks and A. Manitius, "Application of abstract variational theory to hereditary systems - a survey," IEEE Trans. Automat. Control, 19, 524-533 (1974).
    • (1974) IEEE Trans. Automat. Control , vol.19 , pp. 524-533
    • Banks, H.T.1    Manitius, A.2
  • 7
  • 8
  • 9
    • 34347176514 scopus 로고
    • Sur quelques methodes nouvelles dans le calculus des variations
    • N. N. Bogolyubov, "Sur quelques methodes nouvelles dans le calculus des variations," Ann. Math. Pure Appl., 7, 249-271 (1930).
    • (1930) Ann. Math. Pure Appl. , vol.7 , pp. 249-271
    • Bogolyubov, N.N.1
  • 10
    • 12744266723 scopus 로고
    • Local section method in the theory of optimal processes
    • V. G. Boltyanskii, "Local section method in the theory of optimal processes," Differents. Uravn., 4, 2166-2183 (1968);
    • (1968) Differents. Uravn. , vol.4 , pp. 2166-2183
    • Boltyanskii, V.G.1
  • 11
    • 52849093924 scopus 로고
    • English transl.
    • English transl.: in Differential Equations, 4 (1968).
    • (1968) Differential Equations , vol.4
  • 12
    • 0016973122 scopus 로고
    • Optimal solutions to differential inclusions
    • F. H. Clarke, "Optimal solutions to differential inclusions," J. Optim. Theory Appl., 19, 460-478 (1976).
    • (1976) J. Optim. Theory Appl. , vol.19 , pp. 460-478
    • Clarke, F.H.1
  • 13
    • 49449116515 scopus 로고
    • Necessary conditions for a general control problem
    • (D. Russel, Ed.), Academic Press, New York
    • F. H. Clarke, "Necessary conditions for a general control problem," in: Calculus of Variations and Control Theory (D. Russel, Ed.), Academic Press, New York (1976), pp. 257-278.
    • (1976) Calculus of Variations and Control Theory , pp. 257-278
    • Clarke, F.H.1
  • 16
    • 0001563557 scopus 로고
    • Necessary conditions, controllability and the value function for differential-difference inclusons
    • F. H. Clarke and G. G. Watkins, "Necessary conditions, controllability and the value function for differential-difference inclusons," Nonlinear Anal., Theory, Meth., Appl., 10, 1155-1179 (1986).
    • (1986) Nonlinear Anal., Theory, Meth., Appl. , vol.10 , pp. 1155-1179
    • Clarke, F.H.1    Watkins, G.G.2
  • 17
    • 0030182214 scopus 로고    scopus 로고
    • Necessary conditions for functional differential inclusions
    • F. H. Clarke and P. R. Wolenski, "Necessary conditions for functional differential inclusions," Appl. Math. Optim., 34, 51-78 (1996).
    • (1996) Appl. Math. Optim. , vol.34 , pp. 51-78
    • Clarke, F.H.1    Wolenski, P.R.2
  • 19
    • 0026876816 scopus 로고
    • Difference methods for differential inclusions: A survey
    • A. D. Dontchev and F. Lempio, "Difference methods for differential inclusions: a survey," SIAM Rev., 34, 263-294 (1992).
    • (1992) SIAM Rev. , vol.34 , pp. 263-294
    • Dontchev, A.D.1    Lempio, F.2
  • 21
    • 12744256969 scopus 로고
    • Maximum principle for differential inclusions (necessity)
    • R. P. Fedorenko, "Maximum principle for differential inclusions (necessity)," Zh. Vychsl. Mat. Mat. Fiz., 11, 885-893 (1971);
    • (1971) Zh. Vychsl. Mat. Mat. Fiz. , vol.11 , pp. 885-893
    • Fedorenko, R.P.1
  • 23
    • 0023090297 scopus 로고
    • The maximum principle for an optimal solution to differential inclusions with endpoint constraints
    • H. Frankowska, "The maximum principle for an optimal solution to differential inclusions with endpoint constraints," SIAM J. Control Optim., 25, 147-157 (1987).
    • (1987) SIAM J. Control Optim. , vol.25 , pp. 147-157
    • Frankowska, H.1
  • 25
    • 0002365232 scopus 로고    scopus 로고
    • The maximum principle in optimal control of systems governed by semilinear equations
    • (B.S. Mordukhovich and H. J. Sussmann, Eds.), Springer-Verlag, New York
    • B. Ginsburg and A. Ioffe, "The maximum principle in optimal control of systems governed by semilinear equations," in: Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control (B.S. Mordukhovich and H. J. Sussmann, Eds.), Springer-Verlag, New York (1996), pp. 153-202.
    • (1996) Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control , pp. 153-202
    • Ginsburg, B.1    Ioffe, A.2
  • 26
    • 84967717450 scopus 로고
    • Approximate subdifferentials and applications, I: The finite dimensional theory
    • A. D. Ioffe, "Approximate subdifferentials and applications, I: the finite dimensional theory," Trans. Amer. Math. Soc., 281, 289-316 (1984).
    • (1984) Trans. Amer. Math. Soc. , vol.281 , pp. 289-316
    • Ioffe, A.D.1
  • 27
    • 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
  • 28
    • 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 Diff. Eq., 4, 59-87 (1996).
    • (1996) Calc. Var. Partial Diff. Eq. , vol.4 , pp. 59-87
    • Ioffe, A.D.1    Rockafellar, R.T.2
  • 31
    • 0026925456 scopus 로고
    • Lagrange-type extremal trajectories in differential inclusions
    • B. Kaśkosz and S. Lojasiewicz, Jr., "Lagrange-type extremal trajectories in differential inclusions," Syst. Contr. Lett., 19, 241-247 (1992).
    • (1992) Syst. Contr. Lett. , vol.19 , pp. 241-247
    • Kaśkosz, B.1    Lojasiewicz Jr., S.2
  • 32
    • 0038992398 scopus 로고
    • Maximum principle in the theory of optimal time-delayed processes
    • G. L. Kharatishvili, "Maximum principle in the theory of optimal time-delayed processes," Dokl. Akad. Nauk SSSR, 136, 39-42 (1961);
    • (1961) Dokl. Akad. Nauk SSSR , vol.136 , pp. 39-42
    • Kharatishvili, G.L.1
  • 33
    • 52849140078 scopus 로고
    • English transl.
    • English transl.: Sov. Math. Dokl. (1961).
    • (1961) Sov. Math. Dokl.
  • 34
    • 0000966105 scopus 로고    scopus 로고
    • Discrete approximations of differential inclusions
    • F. Lempio and V. Veliov, "Discrete approximations of differential inclusions," Bayrether Math. Schrif., 54, 149-232 (1998).
    • (1998) Bayrether Math. Schrif. , vol.54 , pp. 149-232
    • Lempio, F.1    Veliov, V.2
  • 36
    • 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
  • 37
    • 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
  • 38
    • 0031268176 scopus 로고    scopus 로고
    • Bolza problems with general time constraints
    • P. D. Loewen and R. T. Rockafellar, "Bolza problems with general time constraints," SIAM J. Control Optim., 35, 2050-2069 (1997).
    • (1997) SIAM J. Control Optim. , vol.35 , pp. 2050-2069
    • Loewen, P.D.1    Rockafellar, R.T.2
  • 40
    • 0033078383 scopus 로고    scopus 로고
    • Necessary optimality conditions for differential-difference inclusions
    • L. I. Minchenko, "Necessary optimality conditions for differential-difference inclusions," Nonlinear Anal., Theory Meth. Appl., 35, 307-322 (1999).
    • (1999) Nonlinear Anal., Theory Meth. Appl. , vol.35 , pp. 307-322
    • Minchenko, L.I.1
  • 41
    • 0008830845 scopus 로고
    • Maximum principle in problems of time optimal control with nonsmooth constraints
    • B. S. Mordukhovich, "Maximum principle in problems of time optimal control with nonsmooth constraints," Prikl. Mat. Mekh., 40, 1014-1023 (1976);
    • (1976) Prikl. Mat. Mekh. , vol.40 , pp. 1014-1023
    • Mordukhovich, B.S.1
  • 42
    • 33646940121 scopus 로고
    • English transl.
    • English transl.: J. Appl. Math. Mech., 40, 960-969 (1976).
    • (1976) J. Appl. Math. Mech. , vol.40 , pp. 960-969
  • 43
    • 0008821338 scopus 로고
    • Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems
    • B. S. Mordukhovich, "Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems," Dokl. Akad. Nauk SSSR, 254, 1072-1976 (1980);
    • (1980) Dokl. Akad. Nauk SSSR , vol.254 , pp. 1072-1976
    • Mordukhovich, B.S.1
  • 44
    • 0001254208 scopus 로고
    • English transl.
    • English transl.: Sov. Math. Dokl., 22, 526-530 (1980).
    • (1980) Sov. Math. Dokl. , vol.22 , pp. 526-530
  • 45
    • 5244350789 scopus 로고
    • Optimization and approximation of differential inclusions
    • B. S. Mordukhovich, "Optimization and approximation of differential inclusions," Kibernetika, No. 6, 83-89 (1988);
    • (1988) Kibernetika , Issue.6 , pp. 83-89
    • Mordukhovich, B.S.1
  • 46
    • 4243164141 scopus 로고
    • English transl.
    • English transl.: Cybernetics, 24, 781-787 (1988).
    • (1988) Cybernetics , vol.24 , pp. 781-787
  • 47
    • 52849095140 scopus 로고
    • Approximate maximum principle for finite-difference control systems
    • B. S. Mordukhovich, "Approximate maximum principle for finite-difference control systems," Zh. Vychsl. Mat. Mat. Fiz., 28, 163-177 (1988);
    • (1988) Zh. Vychsl. Mat. Mat. Fiz. , vol.28 , pp. 163-177
    • Mordukhovich, B.S.1
  • 48
    • 38249032546 scopus 로고
    • English transl.
    • English transl.: USSR Comput. Math. Math. Phys., 28, 106-114 (1988).
    • (1988) USSR Comput. Math. Math. Phys. , vol.28 , pp. 106-114
  • 50
    • 0001530657 scopus 로고
    • On variational analysis of differential inclusions
    • (A. Ioffe et al., Eds.), Longman, Harlow
    • B. S. Mordukhovich, "On variational analysis of differential inclusions," in: Optimization and Nonlinear Analysis (A. Ioffe et al., Eds.), Longman, Harlow (1992), pp. 199-213.
    • (1992) Optimization and Nonlinear Analysis , pp. 199-213
    • Mordukhovich, B.S.1
  • 51
    • 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
  • 52
    • 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
  • 53
    • 0000551562 scopus 로고
    • Generalized differential calculus for nonsmooth and set-valued mappings
    • B. S. Mordukhovich, "Generalized differential calculus for nonsmooth and set-valued mappings," J. Math. Anal. Appl., 183, 250-288 (1994).
    • (1994) J. Math. Anal. Appl. , vol.183 , pp. 250-288
    • Mordukhovich, B.S.1
  • 54
    • 84966206068 scopus 로고
    • Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis
    • B. S. Mordukhovich, "Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis," Trans. Amer. Math. Soc., 343, 609-658 (1994).
    • (1994) Trans. Amer. Math. Soc. , vol.343 , pp. 609-658
    • Mordukhovich, B.S.1
  • 55
    • 0028738704 scopus 로고
    • Necessary optimality and controllability conditions for nonsmooth control systems
    • Lake Buena Vista, FL
    • B.S. Mordukhovich, "Necessary optimality and controllability conditions for nonsmooth control systems," in: Proc 33rd Conf. Decision Control, Lake Buena Vista, FL (1994), pp. 3992-3997.
    • (1994) Proc 33rd Conf. Decision Control , pp. 3992-3997
    • Mordukhovich, B.S.1
  • 56
    • 52549095612 scopus 로고    scopus 로고
    • Optimization and finite-difference approximations of differential inclusions with free time
    • (B. S. Mordukhovich and H. J. Sussmann, Eds.), Springer-Verlag, New York
    • B. S. Mordukhovich, "Optimization and finite-difference approximations of differential inclusions with free time," in: Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control (B. S. Mordukhovich and H. J. Sussmann, Eds.), Springer-Verlag, New York (1996), pp. 153-202.
    • (1996) Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control , pp. 153-202
    • Mordukhovich, B.S.1
  • 58
    • 0007633319 scopus 로고    scopus 로고
    • Prox-regular functions in variational analysis
    • R. A. Poliquin and R. T. Rockafellar, "Prox-regular functions in variational analysis," Trans. Amer. Math. Soc., 348, 1805-1838 (1996).
    • (1996) Trans. Amer. Math. Soc. , vol.348 , pp. 1805-1838
    • Poliquin, R.A.1    Rockafellar, R.T.2
  • 59
    • 0001081608 scopus 로고
    • An approach to differentiation of multifunctions and necessary optimality conditions for differential inclusions
    • E. S. Polovinkin and G. V. Smirnov, "An approach to differentiation of multifunctions and necessary optimality conditions for differential inclusions," Differents. Uravn., 22, 944-954 (1986);
    • (1986) Differents. Uravn. , vol.22 , pp. 944-954
    • Polovinkin, E.S.1    Smirnov, G.V.2
  • 60
    • 0010810191 scopus 로고
    • English transl.
    • English transl.: Differential Equations, 22, 660-668 (1986).
    • (1986) Differential Equations , vol.22 , pp. 660-668
  • 63
    • 0014863723 scopus 로고
    • Conjugate convex functions in optimal control and the calculus of variations
    • R. T. Rockafellar, "Conjugate convex functions in optimal control and the calculus of variations," J. Math. Anal. Appl., 32, 174-222 (1970).
    • (1970) J. Math. Anal. Appl. , vol.32 , pp. 174-222
    • Rockafellar, R.T.1
  • 64
    • 84966203576 scopus 로고
    • Existence and duality theorems for convex problems of Bolza
    • R. T. Rockafellar, "Existence and duality theorems for convex problems of Bolza," Trans. Amer. Math. Soc., 159, 1-40 (1971).
    • (1971) Trans. Amer. Math. Soc. , vol.159 , pp. 1-40
    • Rockafellar, R.T.1
  • 65
    • 84966234425 scopus 로고
    • Maximal monotone relations and the second derivatives of nonsmooth functions
    • R. T. Rockafellar, "Maximal monotone relations and the second derivatives of nonsmooth functions," Ann. Inst. H. Poincaré. Anal. Non Linéair, 2, 167-184 (1985).
    • (1985) Ann. Inst. H. Poincaré. Anal. Non Linéair , vol.2 , pp. 167-184
    • Rockafellar, R.T.1
  • 66
    • 0008315455 scopus 로고
    • Dualization of subgradient conditions for optimality
    • R. T. Rockafellar, "Dualization of subgradient conditions for optimality," Nonlinear Anal. Theory, Meth. Appl., 20, 627-646 (1993).
    • (1993) Nonlinear Anal. Theory, Meth. Appl. , vol.20 , pp. 627-646
    • Rockafellar, R.T.1
  • 67
    • 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
  • 69
    • 0025927009 scopus 로고
    • Discrete approximations and optimal solutions of differential inclusions
    • G. V. Smirnov, "Discrete approximations and optimal solutions of differential inclusions," Kibernetika, No. 1, 76-79 (1991);
    • (1991) Kibernetika , Issue.1 , pp. 76-79
    • Smirnov, G.V.1
  • 70
    • 0043028111 scopus 로고
    • English transl.
    • English transl.: Cybernetics, 27, 101-107 (1991).
    • (1991) Cybernetics , vol.27 , pp. 101-107
  • 71
    • 0002671270 scopus 로고    scopus 로고
    • Geometry and optimal control
    • (J. B. Baillieul and J. C. Willems, Eds.), Springer-Verlag, New York
    • H. J. Sussmann, "Geometry and optimal control," in: Mathematical Control Theory (J. B. Baillieul and J. C. Willems, Eds.), Springer-Verlag, New York (1998).
    • (1998) Mathematical Control Theory
    • Sussmann, H.J.1
  • 72
    • 21844524480 scopus 로고
    • Controllability and extremality in nonconvex differential inclusions
    • H. D. Tuan, "Controllability and extremality in nonconvex differential inclusions," J. Optim. Theory Appl., 85, 435-472 (1995).
    • (1995) J. Optim. Theory Appl. , vol.85 , pp. 435-472
    • Tuan, H.D.1
  • 73
    • 0030815471 scopus 로고    scopus 로고
    • The extended Euler-Lagrange condition for nonconvex variational problems
    • R. B. Vinter and H. Zheng, "The extended Euler-Lagrange condition for nonconvex variational problems," SIAM J. Control Optim., 35, 56-77 (1997).
    • (1997) SIAM J. Control Optim. , vol.35 , pp. 56-77
    • Vinter, R.B.1    Zheng, H.2
  • 74
    • 21944450668 scopus 로고    scopus 로고
    • Necessary conditions for optimal control problems with state constraints
    • R. B. Vinter and H. Zheng, "Necessary conditions for optimal control 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
  • 76
    • 0018006591 scopus 로고
    • Controllability of nondifferentiable hereditary processes
    • J. Warga, "Controllability of nondifferentiable hereditary processes," SIAM J. Control Optim., 16, 813-831 (1978).
    • (1978) SIAM J. Control Optim. , vol.16 , pp. 813-831
    • Warga, J.1
  • 77
    • 0020813628 scopus 로고
    • Controllability, extremality, and abnormality in nonsmooth optimal control
    • J. Warga, "Controllability, extremality, and abnormality in nonsmooth optimal control," J. Optim. Theory Appl., 41, 239-260 (1983).
    • (1983) J. Optim. Theory Appl. , vol.41 , pp. 239-260
    • Warga, J.1
  • 78
    • 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. Diff. Equat., 124, 186-204 (1996).
    • (1996) J. Diff. Equat. , vol.124 , pp. 186-204
    • Zhu, Q.J.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.