Skip to main content
SpringerLink
Log in

Sufficient optimality conditions for extremal controls based on functional increment formulas

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We consider the optimal control problem without terminal constraints. With the help of nonstandard functional increment formulas we introduce definitions of strongly extremal controls. Such controls are optimal in linear and quadratic problems. In the general case, the optimality property is provided with an additional concavity condition of Pontryagin’s function with respect to phase variables.

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

    Google Scholar 

  2. Gabasov, R. and Kirillova, F.M. Maximum Principle in Theory of Optimal Control (“Librokom”, Moscow, 2011) [in Russian].

    Google Scholar 

  3. Mangasarian, O. L. “Sufficient Conditions for the Optimal Control of Nonlinear Systems,” SIAM J. Control Optim, No. 4, 139–152 (1966).

    Google Scholar 

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

    Google Scholar 

  5. Clarke, F. Optimization and Nonsmooth Analysis (John Wiley & Sons, New York et al., 1983; Nauka, Moscow, 1988).

    MATH  Google Scholar 

  6. Srochko, V. A. Iterative Methods for Solution of the Optimal Control Problems (Fizmatlit, Moscow, 2000) [in Russian].

    Google Scholar 

  7. Antipina, N. V. and Dykhta, V. A. “Linear Lyapunov-Krotov Functions and Sufficient Conditions for Optimality in the Form of the Maximum Principle,” Russian Mathematics (Iz. VUZ) 46, No. 12, 9–20 (2002).

    MATH  MathSciNet  Google Scholar 

  8. Nikol’skii, M. S. “On the Sufficiency of the Pontryagin Maximum Principle in Some Optimization Problems,” Moscow Univ. Comput. Math. Cybernet. No. 1, 29–37 (2005).

    Google Scholar 

  9. Khailov, E.N. “About Extremal Controls of a Homogeneous Bilinear System Controlled in a Positive Octant,” Works of Steklov’s MIAN 220, 217–235 (1998).

    MathSciNet  Google Scholar 

  10. Swierniak, A. “Cell Cycle as an Object of Control,” J. Biological Systems 3, No. 1, 41–54 (1995).

    Article  Google Scholar 

  11. Srochko, V. A. and Akhmedzhanova, N. S. “Investigation and Solving a Class of Bilinear Problems of Optimal Control,” Vestn. Buryatsk. Univ. Matem. i Inform. 2, 143–148 (2005).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Irkutsk State University, ul. K.Marksa 1, Irkutsk, 664003, Russia

    V. A. Srochko & V. G. Antonik

Authors
  1. V. A. Srochko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. G. Antonik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Srochko.

Additional information

Original Russian Text © V.A. Srochko, V.G. Antonik, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 96–102.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Srochko, V.A., Antonik, V.G. Sufficient optimality conditions for extremal controls based on functional increment formulas. Russ Math. 58, 78–83 (2014). https://doi.org/10.3103/S1066369X14080118

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X14080118

Keywords

Search

Navigation