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Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum. (English. Russian original) Zbl 1183.49003
Russ. Math. 53, No. 1, 1-35 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 1, 3-43 (2009).
Summary: This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton-Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give a survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.

MSC:
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49K15 Optimality conditions for problems involving ordinary differential equations
49K20 Optimality conditions for problems involving partial differential equations
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[1] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961) [in Russian].
[2] L. S. Pontryagin, The Maximum Principle (Fond Matem. Obrazovaniya i Prosveshcheniya, Moscow, 1998) [in Russian].
[3] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974) [in Russian].
[4] V. M. Alekseev, V.M. Tikhomirov, and S. V. Fomin, Optimal Control (Nauka, Moscow, 1979) [in Russian]. · Zbl 0516.49002
[5] F. P. Vasil’ev, Optimization Methods (Faktorial, Moscow, 2002) [in Russian].
[6] R. Gabasov and F. M. Kirillova, The Maximum Principle in the Optimal Control Theory (Nauka i Tekhnika, Minsk, 1974) [in Russian].
[7] A. S. Matveev and V. A. Yakubovich, Optimal Control Systems: Ordinary Differential Equations. Special Problems (Sankt-Peterburgsk. Univ., St.-Petersburg, 2003) [in Russian].
[8] A. V. Arutyunov, G. G. Magaril-Il’yaev, and V. M. Tikhomirov, Pontryagin’s Maximum Principle. Proof and Applications (Faktorial, Moscow, 2006) [in Russian].
[9] A. A. Milyutin, A.V. Dmitruk, and N. P. Osmolovskii, TheMaximum Principle in Optimal Control (Mosk. Gos. Univ., Moscow, 2004) [in Russian].
[10] L. T. Ashchepkov, Lectures in Optimal Control (Dal’nevostochn. Gos. Univ., Vladivostok, 1996) [in Russian].
[11] V. A. Dykhta, ”Lyapunov-Krotov Inequality and Sufficient Conditions in Optimal Control,” J. Math. Sci. 121(2), 2156–2177 (2004). · Zbl 1088.49021 · doi:10.1023/B:JOTH.0000023085.65837.49
[12] V. A. Dykhta, ”Lyapunov-Krotov Inequality and Sufficient Conditions in Optimal Control,” in Itogi Nauki i Tekhn. Sovr. Matem. i ee Prilozh. (VINITI, Moscow, 2006), Vol. 110, pp. 76–108.
[13] N. V. Antipina and V. A. Dykhta, ”Linear Lyapunov-Krotov Functions and Sufficient Conditions of Optimality in the Form of the Maximum Principle,” Izv. Vyssh. Uchebn. Zaved. Mat., 12, 11–21 (2002) [Russian Mathematics (Iz. VUZ) 46 (12), 9–20 (2002)]. · Zbl 1059.49026
[14] A. Ya. Dubovitskii and A. A. Milyutin, ”Theory of the Maximum Principle,” in Methods of the Theory of Extremal Problems in Economics (Nauka, Moscow, 1981), pp. 6–47.
[15] A. P. Afanas’ev, V. V. Dikusar, A. A. Milyutin, and S. A. Chukanov, Necessary Conditions in Optimal Control (Nauka, Moscow, 1990) [in Russian].
[16] A. A. Milyutin, The Maximum Principle in the General Problem of Optimal Control (Fizmatlit, Moscow, 2001) [in Russian]. · Zbl 0998.49003
[17] A.V. Arutyunov, Extremum Conditions. Abnormal and Degenerate Problems (Faktorial, Moscow, 1997) [in Russian].
[18] B. Sh. Mordukhovich, Approximation Methods in Problems of Optimization and Control (Nauka, Moscow, 1988) [in Russian]. · Zbl 0643.49001
[19] F. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983; Nauka, Moscow, 1988). · Zbl 0582.49001
[20] R. B. Vinter, Optimal Control (Burkhauser, Boston-Basel-Berlin, 2000). · Zbl 0952.49001
[21] B. Sh. Mordukhovich, ”Optimal Control of Difference, Differential, and Differential-Difference Inclusions,” in Itogi Nauki i Tekhn. Sovr. Matem. i Ee Prilozh. (VINITI, Moscow, 1999), Vol. 61, pp. 33–65.
[22] S. M. Aseev, ”The Optimal Control Problem for a Differential Inclusion with a Phase Constraint. Smooth Approximations and the Necessary Optimality Conditions” in Itogi Nauki i Tekhn. Sovr. Matem. i Ee Prilozh. (VINITI, Moscow, 1999), Vol. 64, pp. 57–81. · Zbl 0986.49012
[23] F. Clarke, ”The Maximum Principle in Optimal Control: Then and Now,” J. Contr. and Cybern. 34(3), 709–722 (2005). · Zbl 1167.49311
[24] F. Clarke, ”Necessary Conditions in Dynamic Optimization,” Memoirs the Amer. Math. Soc. 173(816), (2005).
[25] A. Arutunov, D. Karamzin, and F. Pereira, ”A Nondegenerate Maximum Principle for Optimal Control Problem with State Constraints,” SIAM J. Contr. Optim. 43(5), 1812–1843 (2005). · Zbl 1116.49013 · doi:10.1137/S0363012903430068
[26] V. V. Dikusar and A. A. Milyutin, Qualitative and Numerical Methods in Maximum Principle (Nauka, Moscow, 1989) [in Russian]. · Zbl 0704.65048
[27] A. A. Milyutin, A. E. Ilyutovich, N. P. Osmolovskii, and S. V. Chukanov, Optimal Control in Linear Systems (Nauka, Moscow, 1993) [in Russian]. · Zbl 0799.93026
[28] V. A. Baturin, V. A. Dykhta, A. I. Moskalenko, et al., Methods for Solving Problems in Control Theory on the Basis of an Extension Principle (Nauka, Novosibirsk, 1990) [in Russian].
[29] A. A. Milyutin, ”Calculus of Variations and Optimal Control,” in Proceedings of the Int. Conf. on the Calculus of Variations and Related topics. Math. Series, Vol. 411, pp. 159–172 (1999). · Zbl 0961.49015
[30] A. A. Milyutin and N. P. Osmolovskii Calculus of Variation and Optimal Control, (Amer.Math. Soc., Providence, Rhode Island, 1998).
[31] V. I. Gurman, The Extension Principle in Control Problems (Fizmatlit, Moscow, 1997) [in Russian]. · Zbl 0905.49001
[32] C. Caratheodory, Calculus of Variations and Partial Differential Equations of the First Order (Holden- Day, San Francisco, 1965), Vol. 2.
[33] L. Yang, Lectures on the Calculus of Variations and Optimal Control Theory, (Saunders Co., Philadelphia, 1969; Mir, Moscow, 1974).
[34] V. F. Krotov, Global Methods in Optimal Control Theory, (Marcel Dekker, New York, 1996). · Zbl 1075.49500
[35] V. F. Krotov and V. I. Gurman, Optimal Control: Methods and Problems, (Nauka, Moscow, 1973) [in Russian].
[36] F. H. Clarke, Yu. S. Ledyaev, and B. J. Stern, ”Proximal Analysis and Feedback Construction,” Trudy Inst. Matem. iMekhan., UrO RAN, Ekaterinburg 6(1–2), 91–109 (2000). · Zbl 1116.93031
[37] F. H. Clarke, Yu. S. Ledyaev, and B. J. Stern, ”Invariance, Monotonicity and Applications,” in Nonlinear Analysis, Differential Equations and Control (NATO ASI, Montreal, 1998; Kluwer Acad. Publ., Dordreht, 1999), pp. 207–305. · Zbl 0936.93001
[38] M.M. Khrustalev, ”Exact Description of the Reachability Sets and Conditions for the Global Optimality of Dynamic Systems, I. Estimates and the Exact Description of Sets of Reachability and Controllability,” Avtomatika i Telemekhan., 5, 62–71 (1988).
[39] M. M. Khrustalev, ”Exact Description of the Reachability Sets and Conditions for the Global Optimality of Dynamic Systems, II. Conditions for the Global Optimality,” Avtomatika i Telemekhan., 7, 70–80 (1988).
[40] F.H. Clarke, ”Nonsmooth Analysis in Control Theory: a Survey,” European J. Control. Fundamental Issues in Control 7(2–3), 145–159 (2001). · Zbl 1293.49033 · doi:10.3166/ejc.7.145-159
[41] F. L. Pereira, ”Control Design for Autonomous Vehicles: a Dynamic Optimization Perspective,” European J. Control. Fundamental Issues in Control 7(2–3), 178–202 (2001). · Zbl 1293.93556 · doi:10.3166/ejc.7.178-202
[42] A. I. Subbotin, Minimax Inequalities and Hamilton-Jacobi Equations (Nauka, Moscow, 1991) [in Russian]. · Zbl 0733.70014
[43] A. I. Subbotin, ”Minimax Inequalities and Hamilton-Jacobi Equations,” in Itogi Nauki i Tekhn. Sovr. Matem. i Ee Prilozh. (VINITI, Moscow, 1999), Vol. 64, pp. 222–231. · Zbl 0986.49018
[44] P. Cannarsa and C. Sinestrari, ”Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control,” in Progress in Nonlinear Differential Equations and Their Applications (Boston: Birkhauser, 2004), Vol. 58. · Zbl 1095.49003
[45] V. N. Ushakov and A. G. Chentsov, ”Andrei Izmailovich Subbotin,” Trudy Inst. Matem. i Mekhan., URO RAN, Ekaterinburg 6(1–2), 3–26 (2000). · Zbl 1137.01326
[46] F. L. Chernous’ko, Estimation of the Phase State of Dynamic Systems. Method of Ellipsoids (Nauka, Moscow, 1988) [in Russian].
[47] F. L. Chernous’ko and N.V. Banichuk, Variational Problems of Mechanics and Control (Numerical Methods) (Nauka, Moscow, 1973) [in Russian].
[48] R. Gabasov and F. M. Kirillova, Qualitative Theory of Optimal Processes (Nauka, Moscow, 1971) [in Russian]. · Zbl 0236.49001
[49] N. E. Kirin, Numerical Methods of the Optimal Control Theory (Leningradsk. Gos. Univ., Leningrad, 1968) [in Russian].
[50] O. V. Vasil’ev, Lectures in Optimization Methods (Irkutsk. Gos. Univ., Irkutsk, 1994) [in Russian].
[51] O. V. Vasil’ev, A. V. Arguchintsev, and V. A. Terletskii, ”Optimization Methods for Systems with Lumped and Distributed Parameters Based on Admissible Variations,” in Proceedings of the 12th Baikal Intern. Conf. ’Optimization Methods and Their Applications’, (Irkutsk, 2001), pp. 52–68.
[52] A. I. Tyatyushkin,Numerical Methods and Program Tools of Optimization of Control Systems (Nauka, Novosibirsk, 1992) [in Russian]. · Zbl 0764.49018
[53] V. A. Srochko, Iterative Methods for Optimal Control Problems (Fizmatlit, MOscow, 2000) [inRussian].
[54] O. V. Vasil’ev, V. A. Dykhta, and V. A. Srochko, ”Optimal Control Problems: the Variational Principle of Maximum and Numerical Solution Methods,” in Nonlinear Control Theory and Its Applications, Ed by V.M. Matrosov, S.N. Vasil’ev, and A. I. Moskalenko (Fizmatlit, Moscow, 2000) [in Russian].
[55] R. P. Fedorenko, Approximate Solution of Optimal Control Problems (Nauka, Moscow, 1978) [in Russian]. · Zbl 0462.49001
[56] F. P. Vasil’ev, Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].
[57] V. F. Dem’yanov and A.M. Rubinov, Approximate Methods of Solving Extremal Problems (Leningradsk. Gos. Univ., Leningrad, 1968) [in Russian].
[58] A. I. Tyatyushkin, Many-Method Technique of Optimization of Control Systems (Nauka, Novosibirsk, 2006) [in Russian].
[59] V. A. Baturin and D. E. Urbanovich, Approximate Optimal Control Methods Based on the Extension Principle (Nauka, Novosibirsk, 1997) [in Russian]. · Zbl 0896.49014
[60] R. Gabasov and F. M. Kirillova, Constructive Optimization Methods. Part 2. Control Problems(Universitetskoe, Minsk, 1984) [in Russian]. · Zbl 0626.90051
[61] R. Gabasov and F. M. Kirillova, ”Real-Time Optimal Control,” in Proceedings of the 2nd Intern. Conf. on Control Problems (Inst. Problem Upravleniya, Moscow, 2003), pp. 20–47.
[62] N. N. Moiseev, Elements of the Theory of Optimal Systems (Nauka, Moscow, 1975) [in Russian]. · Zbl 0314.49005
[63] Yu. G. Evtushenko, Methods for Solving Extremal Problems and Their Application in Optimization Systems (Nauka, Moscow, 1982) [in Russian]. · Zbl 0523.49002
[64] Yu. M. Ermol’ev, V. P. Gulenko, and T. I. Tsarenko, The Finite-Difference Method in Optimal Control Problems (Nauk. Dumka, Kiev, 1978) [in Russian]. · Zbl 0408.49037
[65] T. K. Sirazetdinov, Optimization of Systems with Distributed Parameters (Nauka, Moscow, 1977) [in Russian]. · Zbl 0415.49004
[66] A. I. Egorov, Fundamentals of Control Theory (Fizmatlit, Moscow, 2004) [in Russian].
[67] G. M. Ostrovskii and Yu. M. Volin, Methods for the Optimization of Chemical Reagents (Khimiya, Moscow, 1967) [in Russian].
[68] O. V. Vasil’ev, V. A. Srochko, and V. A. Terletskii, Optimization Methods and Their Applications. Part 2. Optimal Control (Nauka, Novosibirsk, 1990) [in Russian].
[69] V. A. Srochko, ”The Maximum Principle for One Class of Systems with Distributed Parameters,” in Problems of Stability and Optimization of Dynamic Systems (Irkutsk, 1983), pp. 170–182.
[70] V. A. Srochko, ”Optimality Conditions of the Type of the Maximum Principle in Goursat-Darboux Systems,” Sib., Matem. Zhurn. 25(1), 126–133 (1984). · Zbl 0665.49020
[71] V. A. Srochko, The Variational Principle of Maximum and Linearization Methods in Optimal Control Problems (Irkutsk. Gos. Univ., Irkutsk, 1989) [in Russian].
[72] V. A. Terletskii, ”Variational Maximum Principle in Controlled Systems of One-Dimensional Hyperbolic Equations,” Izv. Vyssh. Uchebn. Zaved. Mat., 12, 82–90 (1999) [Russian Mathematics (Iz. VUZ) 43 (12), 78–86 (1999)].
[73] E. P. Bokmelder and V. A. Dykhta, ”To the Theory of the Maximum Principle for Controllable Hyperbolic Systems,” in The Theory and Applied Questions of the Optimal Control (Novosibirsk, 1985), pp. 41–58.
[74] E. P. Bokmelder and V. A. Dykhta, ”The Maximum Principle for Semilinear Hyperbolic Systems under Functional Constraints,” in Differential Equations and Numerical Methods (Nauka, Novosibirsk, 1986), pp. 200–207.
[75] E. P. Bokmelder, V. A. Dykhta, A. I. Moskalenko, and N. A. Ovsyannikova, Extremum Conditions and Constructive Solution Methods for Problems of Optimization of Hyperbolic Systems (Nauka, Novosibirsk, 1993) [in Russian]. · Zbl 0956.49500
[76] A. V. Arguchintsev, ”Non-Classical Optimality Condition in the Problem of Control by Boundary Value Conditions of a Semi-Linear Hyperbolic System,” Izv. Vyssh. Uchebn. Zaved. Mat., 1, 3–11 (1994) [Russian Mathematics (Iz. VUZ) 38 (1), 1–8 (1994)]. · Zbl 0834.49012
[77] A. V. Arguchintsev, ”Solving the Problem of Optimal Control of Initial-Boundary Value Conditions of Hyperbolic System on the Basis of Exact Formulas of Increment,” Izv. Vyssh. Uchebn. Zaved. Mat., 12, 23–29 (2002) [Russian Mathematics (Iz. VUZ) 46 (12), 21–27 (2002].
[78] A. V. Arguchintsev, ”Optimization of Hyperbolic Systems with Controllable Initial Edge Conditions in Form of Differential Relations,” Zhurn. Vychisl. Matem. i Matem. Fiz. 44(2), 285–294 (2004).
[79] A. V. Arguchintsev, Optimal Control of Hyperbolic Systems (Fizmatlit, Moscow, 2007) [in Russian].
[80] V. A. Dykhta, ”The Variational Principle of Maximum and Quadratic Optimality Conditions for Pulse and Singular Processes,” Sib. Matem. Zhurn. 35(1), 70–82 (1994).
[81] V. A. Dykhta, ”The Variational Principle of Maximum for Classical Optimal Control Problems,” Avtomatika i Telemekhan., 4, 47–54 (2002). · Zbl 1073.49012
[82] V.A. Dykhta and O. N. Samsonyuk, Optimal Pulse Control with Applications (Fizmatlit, Moscow, 2003) [in Russian]. · Zbl 1084.49500
[83] L. T. Ashchepkov, Optimal Control of Discontinuous Systems (Nauka, Novosibirsk, 1987) [in Russian]. · Zbl 0688.49016
[84] F. H. Clarke and R. B. Vinter, ”Applications of Optimal Multiprocesses,” SIAM J. Contr. Optim. 27(5), 1048–1071 (1989). · Zbl 0695.49014 · doi:10.1137/0327056
[85] F.H. Clarke and R. B. Vinter, ”Optimal Multiprocesses,” SIAM J. Contr. Optim. 27(5), 1072–1091 (1989). · Zbl 0684.49007 · doi:10.1137/0327057
[86] B. M. Miller and E. Ya. Rabinovich, Optimization of Dynamic Systems with Pulse Controls (Nauka, Moscow, 2005) [in Russian].
[87] A. V. Dmitruk and A. M. Kaganovich, ”The Maximum Principle for Optimal Control Problems with Intermediate Constraints,” in Nonlinear Dynamic Systems and Control (Nauka, Moscow, 2008), Vol. 6. · Zbl 1256.49025
[88] E. S. Levitin, A. A. Milyutin, and N. P. Osmolovskii, ”Conditions of the Higher Orders in Problems with Constraints,” Usp. Mat. Nauk 33(6), 85–147 (1978).
[89] A. V. Arutyunov and R. B. Vinter, ”A Finite-Dimensional Approximation Method in Optimal Control Theory,” Differents. Uravn. 39(11), 1443–1451 (2003). · Zbl 1083.49015
[90] A. V. Arutyunov and R. B. Vinter, ”A Simple ’Finite Approximation’ Proof of the Pontryagin Maximum Principle under Reduced Differentiability Hypotheses,” Set-Valued Analysis 12(1–2), 5–24 (2004). · Zbl 1046.49014 · doi:10.1023/B:SVAN.0000023406.16145.a8
[91] H. J. Sussman, ”Geometry and Optimal Control,” in Mathematical Control Theory, Ed. by J. Baillieul and J. C. Willems (Springer, New York, 1998), pp. 140–198.
[92] H. J. Sussman, ”New Theories of Set-Valued Differentionals and New Versions of the Maximum Principle of Optimal Control Theory,” in Nonlinear Control in the Year 2000, Ed. by A. Isidory, F. Lamanbhi-Lagarrique, and W. Respondek (Springer, London, 2000), pp. 487–526.
[93] A. V. Dmitruk, ”The Maximum Principle for the General Optimal Control Problem with Phase and Mixed Constraints,” in Optimality of Controllable Dynamic Systems (VNIISI, Moscow, 1993), 14, pp. 26–42. · Zbl 1331.49025
[94] A. A. Milyutin, Convex-Valued Lipschitz Differential Inclusions and the Pontryagin Maximum Principle,” in Itogi Nauki i Tekhn. Sovr. Matem. i Ee Prilozh. (VINITI, Moscow, 1999), Vol. 65, pp. 175–184. · Zbl 0986.49013
[95] H. J. Sussmann, ”Needle Variations and Almost Lower Semicontinuous Differential Inclusions,” Set-Valued Analysis 10(2–3), 233–285 (2002). · Zbl 1013.49021 · doi:10.1023/A:1016540217523
[96] S. M. Aseev and A. V. Kryazhimskii, ”The Pontryagin Maximum Principle and Problems of the Optimal Economic Growth,” Trudy MIAN (Nauka, Moscow, 2007), Vol. 257.
[97] A. V. Arutyunov, ”The Pontryagin Maximum Principle and Sufficient Conditions for Nonlinear Problems,” Differents. Uravn. 39(12), 1587–1595 (2003).
[98] Kh. G. Guseinov and V. N. Ushakov, ”Strongly and Weakly Invariant Sets with Respect to Differential Inclusion, Their Derivatives, and Application to Control Problems,” Differents. Uravn. 26(11), 1888–1894 (1990).
[99] T. Donchev, V. Rios, and P. Wolenski, ”Strong Invariance and One-Sided Lipschitz Multifunctions,” Nonlinear Analysis. TMA 60(5), 849–862 (2005). · Zbl 1068.34010 · doi:10.1016/j.na.2004.09.050
[100] J.-P. Aubin and A. Cellina, Differential Inclusions (Springer-Verlag, Berlin, 1984). · Zbl 0538.34007
[101] A. B. Kurzhanskii and T. F. Filippova, ”On the Theory of Trajectory Tubes–a Mathematical Formalism for Uncertain Dynamics, Viability and Control,” in Advanced in Nonlinear Dynamics and Control: a Report from Russia, Ed by A. B. Kurzhanskii (Burkhauser, Boston, 1993), pp. 122–188. · Zbl 0912.93040
[102] A.I. Moskalenko, Methods of Nonlinear Mappings in Optimal Control (Nauka, Novosibirsk, 1983) [in Russian]. · Zbl 0544.92015
[103] V. A. Dykhta, ”The General Scheme of Transforms of Extremal Problems and Its Applications in Optimal Control,” in Integrodifferential Equations and Their Applications (Irkutsk, 1987), pp. 82–91.
[104] V. A. Dykhta and N. V. Antipina, ”A Sufficient Optimality Condition for Pulse Control Problems,” Izv. Ross. Akad. Nauk. Teoriya i Sistemy Upravleniya, 4, 76–83 (2004).
[105] A. A. Milyutin, ”An Example of an Optimal Control Problem Whose Extremals Possess a Continual Set of Discontinuities of the Control Function,” Russian J. Math. Physics 1(3), 397–402 (1994). · Zbl 0915.49018
[106] A. B. Kurzhanskii and I. Valyi, Ellipsoidal Calculus for Estimation and Control (Birkhauser, Boston, 1997).
[107] F. H. Clarke, ”A Proximal Characterization of the Reachable Set,” Systemand Control Letters 27, 195–197 (1996). · Zbl 0866.93008 · doi:10.1016/0167-6911(95)00056-9
[108] R. B. Vinter, ”A Characterization of the Reachable Set for Nonlinear Control System,” SIAM J. Contr. Optim. 18(6), 599–610 (1980). · Zbl 0457.93013 · doi:10.1137/0318044
[109] P. Cannarsa and H. Frankowska, ”Some Characterizations of Optimal Trajectories in Control Theory,” SIAM J. Contr. Optim. 29(6), 1322–1347 (1991). · Zbl 0744.49011 · doi:10.1137/0329068
[110] P. Cannarsa, H. Frankowska, and S. Sinestrari, ”Optimality Conditions and Synthesis for the Minimum Time Problem,” Set-Valued Analysis 8(1–2), 127–148 (2000). · Zbl 0988.49010 · doi:10.1023/A:1008726610555
[111] P. Wolenski and Yu. Shuang, ”Proximal Analysis and the Minimal Time Function,” SIAM J. Contr. Optim. 36(3), 1048–1072 (1998). · Zbl 0930.49016 · doi:10.1137/S0363012996299338
[112] M. M. Khrustalev, ”The Necessary and Sufficient Optimality Conditions in the Form of the Bellman Equation,” Sov. Phys. Dokl. 242(5), 1023–1026 (1978).
[113] M. Motta and F. Rampazzo, ”Dynamic Programming for Nonlinear System Driven by Ordinary and Impulsive Controls,” SIAM J. Contr. Optim. 34(1), 199–225 (1996). · Zbl 0843.49021 · doi:10.1137/S036301299325493X
[114] A. V. Stefanova, ”The Hamilton-Jacobi-Bellman Equation in a Nonlinear Pulse Control Problem,” Trudy Inst. Matem. i Mekhan., URORAN, Ekaterinburg 5, pp. 301–318 (1998). · Zbl 1007.49013
[115] F. L. Pereira and A. S. Matos, ”Hamilton-Jacobs Conditions for a Measure Differential Inclusion Control Problem,” in Proceedings of the 12th Baikal Internat. Conf. ’Optimization Methods and Their Applications’, (Irkutsk, 2001), pp. 237–245.
[116] R. B. Vinter, ”Weakest Conditions for Existence of Lipschitz Continuous Krotov Functions in Optimal Control Theory,” SIAM J. Contr. Optim. 21(2), 215–234 (1983). · Zbl 0507.49016 · doi:10.1137/0321012
[117] R. B. Vinter, ”New Global Optimality Conditions in Optimal Control Theory,” SIAM J. Contr. Optim. 21(2), 235–245 (1983). · Zbl 0507.49017 · doi:10.1137/0321013
[118] J.-P. Aubin and H. Frankowska, Set-Valued Analysis (Birkhauser, Boston-Basel-Berlin, 1990). · Zbl 0713.49021
[119] N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].
[120] I. A. Krylov and F. L. Chernous’ko, ”The Method of Successive Approximations for Solving Optimal Control Problems,” Zhurn. Vychisl. Matem. i Matem. Fiz. 2(6), 1132–1139 (1962).
[121] I. A. Krylov and F.L. Chernous’ko, ”An Algorithm of the Method of Successive Approximations for Optimal Control Problems,” Zhurn. Vychisl. Matem. i Matem. Fiz. 12(1), 14–34 (1972).
[122] H. J. Kelly, R. E. Kopp, and H. G. Moyer, ”Successive Approximation Techniques for Trajectory Optimization,” in Proceedings of Symp. on Vehicle System Optimization (New York, 1961), pp. 360–391.
[123] V. V. Velichenko, ”A Numerical Method for Solving Optimal Control Problems,” Zhurn. Vychisl. Matem. i Matem. Fiz. 6(4), 635–647 (1966).
[124] O. V. Vasil’ev and A. I. Tyatyushkin, ”On One Method for Solving Optimal Control Problems Based on the Maximum Principle,” Zhurn. Vychisl. Matem. i Matem. Fiz. 21(6), 1376–1384 (1981). · Zbl 0482.49018
[125] N. V. Tarasenko, ”One Method for Solving Optimal Control Problems Based on the Integral Maximum Principle,” in Discrete and Distributed Systems (Irkutsk, 1981), pp. 142–150.
[126] A. A. Lyubushin and F. L. Chernous’ko, ”Method of Successive Approximations for Calculating an Optimal Control,” Izv. Akad. Nauk SSSR, Tekhn. Kibernetika, 2, 147–159 (1983).
[127] O. V. Vasiliev, Optimization Methods (World Federation Publishers Company Inc., Atlanta, 1996). · Zbl 0883.49001
[128] O. V. Vasil’ev, ”A Remark to the Algorithm of Successive Approximations Based on the Maximum Principle,” in Differential and Integral Equations (Irkutsk, 1980), pp. 167–178.
[129] V. G. Antonik and V. A. Srochko, ”Solution of Optimal Control Problems by Linearization Methods,” Zhurn. Vychisl. Matem. i Matem. Fiz. 32(7), 979–991 (1992). · Zbl 0784.49001
[130] V. G. Antonik and V. A. Srochko, ”The Projection Method in Linear-Quadratic Problems of Optimal Control,” Zhurn. Vychisl. Matem. i Matem. Fiz. 38(4), 564–572 (1998). · Zbl 0953.49032
[131] V. S. Zakharchenko and V. A. Srochko, ”The Method of Increments for Solving Quadratic Problems of Optimal Control,” Izv. Ross. Akad. Nauk. Teoriya i Sistemy Upravleniya, 6, 145–154 (1995).
[132] N. V. Mamonova and V. A. Srochko, ”Iteration Procedures for Solving Problems of Optimal Control on the Basis of Quasigradient Approximations,” Izv. Vyssh. Uchebn. Zaved. Mat., 12, 55–67 (2001) [Russian Mathematics (Iz. VUZ) 45 (12), 52–64 (2001)]. · Zbl 1057.49022
[133] V. A. Srochko, ”Modernization of Gradient Type Methods in Optimal Control Problems,” Izv. Vyssh. Uchebn. Zaved. Mat., 12, 66–78 (2002) [Russian Mathematics (Iz. VUZ) 46 (12), 64–76 (2002)].
[134] V. I. Boldyrev, ”Numerical Solution of Optimal Control Problems,” Izv. Ross. Akad. Nauk. Teoriya i Sistemy Upravleniya, 3, 85–92 (2000).
[135] V. A. Baturin, ”Approximate Methods for Solving Optimal Control Problems on the Base of V. F. Krotov Sufficient Optimality Conditions,” in Proceedings of the IXth Intern. Conf. ’Analytic Mechanics, Stability, and Motion Control’ dedicated to the 105th anniversary of N. G. Chetaev. Part 3. Control and Optimization (Irkutsk, 2007), pp. 30–47.
[136] S. N. Avvakumov, Yu. N. Kiselev, and M. V. Orlov, ”Methods for Solving Optimal Control Problems on the Base of the Pontryagin Maximum Principle,” Trudy Matem. Inst. Ross. Akad. Nauk 211, 3–31 (1995). · Zbl 0888.49020
[137] A. S. Strekalovskii, Elements of Nonconvex Optimization (Nauka, Novosibirsk, 2003) [in Russian]. · Zbl 1103.26012
[138] N. V. Balashevich, R. Gabasov, and F. M. Kirillova, ”Numerical Methods for Program and Positional Optimization of Piecewise Linear Systems,” Zhurn. Vychisl. Matem. i Matem. Fiz. 41(11), 1658–1674 (2001). · Zbl 1032.49041
[139] R. Gabasov, F.M. Kirillova, and T. G. Khomitskaya, ”Program and Positional Solutions of a Terminal Linear-Convex Optimal Control Problem,” Izv. Vyssh. Uchebn. Zaved. Mat., 12, 3–16 (2004) [Russian Mathematics (Iz. VUZ) 48 (12), 1–14 (2004)].
[140] S. L. Kaganovich, ”Optimization of Linear Systems with Variable Structure,” Zhurn. Vychisl. Matem. i Matem. Fiz. 21(2), 306–314 (1981). · Zbl 0472.49020
[141] V. F. Krotov and I. N. Fel’dman, ”An Iterative Solution Method for the Optimal Control Problems,” Izv. Akad. Nauk SSSR. Tekhn. Kibernetika, 2, 160–168 (1983).
[142] R. Gabasov and F.M. Kirillova, Singular Optimal Controls (Nauka, Moscow, 1973) [in Russian]. · Zbl 1177.93037
[143] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Nauka, Moscow, 1985) [in Russian]. · Zbl 0571.34001
[144] L. I. Rozonoer, ”The Maximum Principle of L. S. Pontryagin in the Theory of Optimal Systems. I,” Avtomatika i Telemekhan. 20(10), 1320–1334 (1959).
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