Skip to main content
SpringerLink
Log in

Optimal processes in smooth-convex minimization problems

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The paper elaborates a general method for studying smooth-convex conditional minimization problems that allows one to obtain necessary conditions for solutions of these problems in the case where the image of the mapping corresponding to the constraints of the problem considered can be of infinite codimension.

On the basis of the elaborated method, the author proves necessary optimality conditions in the form of an analog of the Pontryagin maximum principle in various classes of quasilinear optimal control problems with mixed constraints; moreover, the author succeeds in preserving a unified approach to obtaining necessary optimality conditions for control systems without delays, as well as for systems with incommensurable delays in state coordinates and control parameters. The obtained necessary optimality conditions are of a constructive character, which allows one to construct optimal processes in practical problems (from biology, economics, social sciences, electric technology, metallurgy, etc.), in which it is necessary to take into account an interrelation between the control parameters and the state coordinates of the control object considered. The result referring to systems with aftereffect allows one to successfully study many-branch product processes, in particular, processes with constraints of the “bottle-neck” type, which were considered by R. Bellman, and also those modern problems of flight dynamics, space navigation, building, etc. in which, along with mixed constraints, it is necessary to take into account the delay effect.

The author suggests a general scheme for studying optimal process with free right endpoint based on the application of the obtained necessary optimality conditions, which allows one to find optimal processes in those control systems in which no singular cases arise.

The author gives an effective procedure for studying the singular case (the procedure for calculating a singular control in quasilinear systems with mixed constraints.

Using the obtained necessary optimality conditions, the author constructs optimal processes in concrete control systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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).

    Google Scholar 

  2. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  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.

    Google Scholar 

  4. H. T. Banks, “Necessary conditions for control problems with variable time lags,” SIAM J. Control, 6, 9–47 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Bellman, Dynamical Programming [Russian translation], IL, Moscow (1960).

    Google Scholar 

  6. L. Bittner, “New conditions for validity of the Lagrange multiplier rule,” Math. Nachrichten, 48, 353–370 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. G. Boltyanskii, Mathematical Methods of Optimal Control [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  8. V. F. Demyanov, “On necessary extremum conditions in problems with aftereffect,” Dokl. Akad. Nauk SSSR, 166, No. 2, 275–277 (1966).

    MathSciNet  Google Scholar 

  9. A. Ya. Dubovitskii and A. A. Milyutin, “Extremum problems with constraints,” Dokl. Akad. Nauk SSSR, 149, No. 4, 459–462 (1963).

    Google Scholar 

  10. R. Gabasov, “On the theory of necessary optimality conditions for singular controls,” Dokl. Akad. Nauk SSSR, 183, No. 2, 252–255 (1968).

    MathSciNet  Google Scholar 

  11. R. Gabasov and F. M. Kirillova, “Maximum principle for Pontryagin extremals,” Differents. Uravn., 4, No. 6, 963–972 (1968).

    MATH  MathSciNet  Google Scholar 

  12. R. Gabasov and F. M. Kirillova, Singular Optimal Controls [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  14. T. Guinn, “Reduction of delayed optimal control problems to non-delayed problems,” J. Optim. Theory Appl., 18, No. 3, 371–377 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  15. V. I. Gurman, “Sufficient optimality conditions,” in: Optimal Control [in Russian], Moscow (1978), pp. 53–68.

  16. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  17. G. L. Kharatishvili, “Maximum principle in the theory of optimal processes with delays,” Dokl. Akad. Nauk SSSR, 136, No. 1, 39–42 (1961).

    Google Scholar 

  18. G. L. Kharatishvili, “Maximum principle in extremal problems with delays,” Tr. TGU, 129, 149–156 (1968).

    Google Scholar 

  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).

    Google Scholar 

  20. G. L. Kharatishvili and T. A. Tadumadze, “Nonlinear optimal systems with variable delays,” Mat. Sb., 107(149), No. 4 (12), 613–633 (1978).

    MathSciNet  Google Scholar 

  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.

    Google Scholar 

  23. V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  24. E. B. Lee and L. Markus, Foundations of Optimal Control Theory [Russian translation], Nauka, Moscow (1972).

    MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  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.

  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).

    Google Scholar 

  29. M. D. Mardanov, Some Problems of the Mathematical Theory of Optimal Processes with Delays [in Russian], Azerbadjan University, Baku (1987).

    Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  32. T. K. Melikov, Singular Controls in Systems with Aftereffect [in Russian], ELM, Baku (2002).

    Google Scholar 

  33. L. W. Neustadt, Optimization: A Theory of Necessary Conditions, Princeton Univ. Press, Princeton (1976).

    MATH  Google Scholar 

  34. N. M. Ogustoreli, Time-Delay Control Systems, Academic Press, New York-London (1966).

    Google Scholar 

  35. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  36. L. I. Rozonoer, “L. S. Pontryagin maximum principle in optimal system theory,” Avtomat. Telemekh., 20, No. 12, 1561–1578 (1959).

    MathSciNet  Google Scholar 

  37. L. Schwartz, Analysis, Vol. 1 [Russian translation], Mir, Moscow (1968).

    Google Scholar 

  38. T. A. Tadumadze, Certain Problems of Qualitative Optimal Control Theory [in Russian], Tbilisi Univ., Tbilisi (1983).

    Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  40. Z. A. Tsintsadze, “Calculation of a singular control in an optimal problem with mixed constraints,” Reihe Mathematik, Greifswald 13, 59–63 (1985).

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  44. Z. A. Tsintsadze, “Extremality principle in a linear optimal control problem with mixed constraints,” Trudy IPM TGU, No. 41, 149–155 (1991).

  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).

    MATH  MathSciNet  Google Scholar 

  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).

    Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  51. Z. Tsintsadze, “On one problem of synthesis of optimal controls,” Repts. Vekua Inst. Appl. Math., 15, No. 2, 71–74 (2000).

    MathSciNet  Google Scholar 

  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).

    Google Scholar 

  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).

  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).

    MathSciNet  Google Scholar 

  56. Z. Tsintsadze, “Optimal processes in the specific control systems,” Mem. Diff. Equat. Math. Phys., 31, 135–138 (2004).

    MATH  MathSciNet  Google Scholar 

  57. K. Sh. Tsiskaridze, “Extremal problems in Banach spaces,” in: Some Problems of Mathematical Optimal Control Theory, Tbilisi Univ., Tbilisi (1975), pp. 3–150.

    Google Scholar 

  58. F. P. Vasil’ev, Numerical Methods for Solving Extremal Problems [in Russian], Nauka, Moscow (1988).

    MATH  Google Scholar 

  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.

  60. S. A. Zinzadse, “Bemerkungen zu einem sonderfall glatt-convexen Optimierungs-aufgaben,” J. Optimization, 2, 191–194 (1988).

    Article  MathSciNet  Google Scholar 

  61. S. A. Zinzadse, “Notwendige Optimalitatsbedignungen in einigem glatt-convexen Minimiezirungsaufgaben,” Reihe Math., 22, 95–98 (1989).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. I. Dzhavakhishvili Tbilisi State University, Tbilisi

    Z. A. Tsintsadze

Authors
  1. Z. A. Tsintsadze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. A. Tsintsadze.

Additional information

__________

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 42, Optimal Control, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tsintsadze, Z.A. Optimal processes in smooth-convex minimization problems. J Math Sci 148, 399–480 (2008). https://doi.org/10.1007/s10958-008-0011-6

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-0011-6

Keywords

Search

Navigation