Optimal processes in smooth-convex minimization problems. (English. Russian original) Zbl 1194.49031
J. Math. Sci., New York 148, No. 3, 399-480 (2008); translation from Sovrem. Mat. Prilozh. 42, 100-170 (2006).
Using a separation theorem first a Lagrange constraint elimination principle is proved followed by an extremality principle in Banach spaces, where convexity, differentiability and regularity conditions must hold. The author shows that the conditional minimization problem considered in this paper is a special case of the general smooth-convex problems studied in [A. A. Ioffe and V. M. Tikhomirov, Theorie der Extremalprobleme. Nelineinyi analiz i ego prilozenija. Moskau: Verlag ‘Nauka’, Hauptredaktion für physikalisch-mathematische Literatur. (1974; Zbl 0292.90042)], for which the regularity conditions formulated there do not hold (infinite codimensions are allowed). After the introducing section there follow three special sections having the titles:
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- Necessary optimality conditions in optimal control problems with mixed constraints and convexity conditions,
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- Necessary optimality conditions for processes with aftereffect,
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- Applications to concrete problems.
Reviewer: Alfred Göpfert (Halle)
MSC:
| 49K27 | Optimality conditions for problems in abstract spaces |
| 49N90 | Applications of optimal control and differential games |
Keywords:
optimal process; process with aftereffect (delay); bottleneck processes; infinite codimension; Lagrange constraint elimination principleCitations:
Zbl 0292.90042References:
| [1] | A. P. Afanas’ev, V. V. Dikusar, A. A. Milyutin, and S. A. Chukanov, Necessary Conditions in Optimal Control [in Russian], Nauka, Moscow (1990). · Zbl 0724.49014 |
| [2] | V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979). · Zbl 0516.49002 |
| [3] | A. V. Arutyunov, ”Perturbations of constrained extremal problems and necessary optimality conditions,” in: Progress in Science and Technology, Series on Mathematical Analysis [in Russian], Vol. 27, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (1989), pp. 147–235. |
| [4] | H. T. Banks, ”Necessary conditions for control problems with variable time lags,” SIAM J. Control, 6, 9–47 (1968). · Zbl 0159.13002 · doi:10.1137/0306002 |
| [5] | R. Bellman, Dynamical Programming [Russian translation], IL, Moscow (1960). |
| [6] | L. Bittner, ”New conditions for validity of the Lagrange multiplier rule,” Math. Nachrichten, 48, 353–370 (1971). · Zbl 0232.49003 · doi:10.1002/mana.19710480127 |
| [7] | V. G. Boltyanskii, Mathematical Methods of Optimal Control [in Russian], Nauka, Moscow (1969). · Zbl 0249.49010 |
| [8] | V. F. Demyanov, ”On necessary extremum conditions in problems with aftereffect,” Dokl. Akad. Nauk SSSR, 166, No. 2, 275–277 (1966). |
| [9] | A. Ya. Dubovitskii and A. A. Milyutin, ”Extremum problems with constraints,” Dokl. Akad. Nauk SSSR, 149, No. 4, 459–462 (1963). |
| [10] | R. Gabasov, ”On the theory of necessary optimality conditions for singular controls,” Dokl. Akad. Nauk SSSR, 183, No. 2, 252–255 (1968). · Zbl 0185.23902 |
| [11] | R. Gabasov and F. M. Kirillova, ”Maximum principle for Pontryagin extremals,” Differents. Uravn., 4, No. 6, 963–972 (1968). |
| [12] | R. Gabasov and F. M. Kirillova, Singular Optimal Controls [in Russian], Nauka, Moscow (1973). · Zbl 1177.93037 |
| [13] | R. V. Gamkrelidze and G. L. Kharatishvili, ”Extremal problems in linear topological spaces,” Izv. Akad. Nauk SSSR, Ser. Mat., 33, No. 4, 781–839 (1969). · Zbl 0182.18402 |
| [14] | T. Guinn, ”Reduction of delayed optimal control problems to non-delayed problems,” J. Optim. Theory Appl., 18, No. 3, 371–377 (1976). · Zbl 0304.49017 · doi:10.1007/BF00933818 |
| [15] | V. I. Gurman, ”Sufficient optimality conditions,” in: Optimal Control [in Russian], Moscow (1978), pp. 53–68. · Zbl 0422.49018 |
| [16] | A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974). |
| [17] | G. L. Kharatishvili, ”Maximum principle in the theory of optimal processes with delays,” Dokl. Akad. Nauk SSSR, 136, No. 1, 39–42 (1961). |
| [18] | G. L. Kharatishvili, ”Maximum principle in extremal problems with delays,” Tr. TGU, 129, 149–156 (1968). |
| [19] | G. L. Kharatishvili, Z. A. Macaidze, N. I. Markozashvili, and T. A. Tadumadze, Abstract Variational Theory and Its Applications to Optimal Problems with Delays [in Russian], Metsniereba, Tbilisi (1973). |
| [20] | G. L. Kharatishvili and T. A. Tadumadze, ”Nonlinear optimal systems with variable delays,” Mat. Sb., 107(149), No. 4 (12), 613–633 (1978). |
| [21] | R. Klotzler, Mehrdimensionale Variationsrtchnung, Leipzig (1969). |
| [22] | N. N. Krasovskii, ”Optimal processes in systems with delay,” in: Optimal Systems, Statistical Methods, Proceeding of IInd IFAC Congress [in Russian], Vol. 2, Nauka, Moscow (1965), pp. 179–244. |
| [23] | V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control [in Russian], Nauka, Moscow (1973). |
| [24] | E. B. Lee and L. Markus, Foundations of Optimal Control Theory [Russian translation], Nauka, Moscow (1972). · Zbl 0247.49001 |
| [25] | Z. A. Machaidze, ”On the uniqueness problem of an optimal control for systems with delays,” Soobshch. Akad. Nauk GSSR, 78, No. 2, 285–288 (1975). |
| [26] | A. Manitius, ”Optimal control of hereditary systems in control theory and topics in functional analysis,” in: Int. Atom. Energy Ag., Vienna (1976), pp. 43–178. · Zbl 0353.49023 |
| [27] | K. B. Mansimov, ”Some problems of the qualitative theory of optimal processes. Part I. Ordinary continuous systems,” Deposited at VINITI, No. 3118-82, Baku (1982). |
| [28] | K. B. Mansimov, Singular Controls in Systems With Delay [in Russian], ELM, Baku (1999). · Zbl 0449.49014 |
| [29] | M. D. Mardanov, Some Problems of the Mathematical Theory of Optimal Processes with Delays [in Russian], Azerbadjan University, Baku (1987). · Zbl 0825.34039 |
| [30] | N. I. Markozashvili, ”Necessary optimality conditions for control systems with previous history,” in: Some Problems of Mathematical Optimal Control Theory [in Russian], Tbilisi Univ., Tbilisi (1975), pp. 151–180. |
| [31] | A. S. Matveev, ”Optimal control problems with delays of the general form and state constraints, Izv. Akad. Nauk SSSR, Ser. Mat., 52, No. 6, 1200–1229 (1988). |
| [32] | T. K. Melikov, Singular Controls in Systems with Aftereffect [in Russian], ELM, Baku (2002). |
| [33] | L. W. Neustadt, Optimization: A Theory of Necessary Conditions, Princeton Univ. Press, Princeton (1976). · Zbl 0353.49003 |
| [34] | N. M. Ogustoreli, Time-Delay Control Systems, Academic Press, New York-London (1966). |
| [35] | L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Fizmatgiz, Moscow (1961). |
| [36] | L. I. Rozonoer, ”L. S. Pontryagin maximum principle in optimal system theory,” Avtomat. Telemekh., 20, No. 12, 1561–1578 (1959). |
| [37] | L. Schwartz, Analysis, Vol. 1 [Russian translation], Mir, Moscow (1968). |
| [38] | T. A. Tadumadze, Certain Problems of Qualitative Optimal Control Theory [in Russian], Tbilisi Univ., Tbilisi (1983). · Zbl 0579.49003 |
| [39] | T. Tadumadze and L. Alkhazishvili, ”Necessary conditions of optimality for optimal problems with delays and a discontinuous initial condition,” Mem. Diff. Equat. Math. Phys., 22, 154–158 (2001). · Zbl 0984.49014 |
| [40] | Z. A. Tsintsadze, ”Calculation of a singular control in an optimal problem with mixed constraints,” Reihe Mathematik, Greifswald 13, 59–63 (1985). |
| [41] | Z. A. Tsintsadze, ”Calculation of singular controls in optimal problems with mixed constraints,” Optimal Problems in Variable-Structure Systems [in Russian], Tbilisi Univ., Tbilisi (1985), pp. 155–175. |
| [42] | Z. A. Tsintsadze, ”Necessary extremality conditions in certain smooth-convex problems,” in: Optimal Control, Proceedings of the Institute of Applied Mathemetics, Tbilisi State University, No. 27 [in Russian], Tbilisi Univ., Tbilisi (1988), pp. 138–162. · Zbl 0736.49022 |
| [43] | Z. A. Tsintsadze, ”Necessary optimality conditions for a certain class of convex control problems with mixed constraints,” in: Extended Session Reports of the Institute of Applied Mathematics Workshop, Tbilisi State University Vol. 3, No. 3 [in Russian], Tbilisi (1988), pp. 181–184. |
| [44] | Z. A. Tsintsadze, ”Extremality principle in a linear optimal control problem with mixed constraints,” Trudy IPM TGU, No. 41, 149–155 (1991). · Zbl 1012.49017 |
| [45] | Z. A. Tsintsadze, ”Extremality principle in a linear optimal control problem with isoperimetric and mixed constraints,” J. Appl. Math. Inform., 1, No. 1, 153–157 (1996). · Zbl 1055.49514 |
| [46] | Z. Tsintsadze, ”Necessary conditions for extremality for nonclassical quasi-linear problems of the calculus of variations,” Repts. Vekua Inst. Appl. Math., 11, Nos. 1–3, 35–38 (1996). |
| [47] | Z. Tsintsadze, ”The Lagrange principle of taking restrictions off and applications to linear optimal control problems in the presence of mixed restrictions and delays,” Mem. Diff. Equat. Math. Phys., 11, 105–128 (1997). · Zbl 0906.49015 |
| [48] | Z. Tsintsadze, ”Necessary conditions of optimality for autonomous quasi-linear control systems with mixed restrictions,” in: Proc. Int. Symp. Dedicated to the 90th Birthday of Acad. I. Vekua, Metsniereba, Tbilisi (1997), p. 47. |
| [49] | Z. Tsintsadze, ”The maximum principle for quasi-linear systems with a deviating argument of neutral type in the presence of mixed restrictions,” in: Proc. Symp. Dedicated to the 95th Birthday of Acad. V. Kupradze, Tbilisi (1998), p. 30. |
| [50] | Z. Tsintsadze, ”Necessary conditions of optimality of processes with parameters to quasi-linear problems of control in the presence of mixed restrictions,” Repts. Vekua Inst. Appl. Math., 13, Nos. 2–3, 48–52 (1998). |
| [51] | Z. Tsintsadze, ”On one problem of synthesis of optimal controls,” Repts. Vekua Inst. Appl. Math., 15, No. 2, 71–74 (2000). · Zbl 1229.49037 |
| [52] | Z. A. Tsintsadze, ”Calculation of a singular optimal control in quasilinear control systems with mixed constraints,” Proc. Javakhishvili Tbilisi State University, AMI, 342, 55–60 (2000). · Zbl 1171.49305 |
| [53] | Z. Tsintsadze, ”Optimization of linear controlled systems with restrictions of ’narrow place’ type,” in: Proc. Symp. on DEMPH, Tbilisi (2001), http://www.rmi.acnet.ge/DEMPH |
| [54] | Z. Tsintsadze, ”Optimization of linear systems with restrictions of ’narrow place’ type,” in: Proc. 6th World Multi-Conference on SCI, Orlando, Florida, SCI 2002/ISAS (2002). · Zbl 1065.49017 |
| [55] | Z. Tsintsadze, ”The optimal process in the linear controlled systems under restrictions of the ”narrow place” type,” Repts. Vekua Inst. Appl. Math., 17, No. 2, 53–57 (2002). · Zbl 1065.49017 |
| [56] | Z. Tsintsadze, ”Optimal processes in the specific control systems,” Mem. Diff. Equat. Math. Phys., 31, 135–138 (2004). · Zbl 1085.49033 |
| [57] | K. Sh. Tsiskaridze, ”Extremal problems in Banach spaces,” in: Some Problems of Mathematical Optimal Control Theory, Tbilisi Univ., Tbilisi (1975), pp. 3–150. |
| [58] | F. P. Vasil’ev, Numerical Methods for Solving Extremal Problems [in Russian], Nauka, Moscow (1988). |
| [59] | M. I. Zelikin and V. F. Borisov, ”Singular optimal regimes in problems of mathematical economics,” in: Contemporary Mathematics and Its Application, Vol. 11, Optimal Control, Tbilisi (2003), pp. 3–161. · Zbl 1088.49022 |
| [60] | S. A. Zinzadse, ”Bemerkungen zu einem sonderfall glatt-convexen Optimierungs-aufgaben,” J. Optimization, 2, 191–194 (1988). · Zbl 0646.49015 · doi:10.1080/02331938808843336 |
| [61] | S. A. Zinzadse, ”Notwendige Optimalitatsbedignungen in einigem glatt-convexen Minimiezirungsaufgaben,” Reihe Math., 22, 95–98 (1989). |
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