Advertisement

Extension technology and extrema selections in a stochastic multistart algorithm for optimal control problems

  • Alexander Yu. GornovEmail author
  • Tatiana S. Zarodnyuk
  • Anton S. Anikin
  • Evgeniya A. Finkelstein
Article
  • 13 Downloads

Abstract

The paper proposes a method for finding the minimum value of a functional in nonlinear nonconvex optimal control problems. The method takes advantage of the hidden convexity property of the controlled differential equations systems. Application of the multistart idea with extrema selection procedures makes it possible to create software that does not strongly depend on the problem size and supplies additional information about the object under investigation. Three test problems are considered to show specific properties of using the stochastic multistart algorithm and extension numerical technology.

Keywords

Optimal control Hidden convexity Global extremum Nonlinear system 

Notes

References

  1. 1.
    Bondarenko, A.S., Bortz, D.M., More J.J.: COPS: large-scale nonlinearly constrained optimization problems. Technical Memorandum ANL/MCS-TM-237, pp. 12–14 (1999)Google Scholar
  2. 2.
    Evtushenko, Y.G., Polovinkin, M.A.: The parallel methods for solving global optimization problems. In: Proceeding of IV International Conference on Parallel computing and control problems, pp. 18–39 (2008). (in Russian)Google Scholar
  3. 3.
    Floudas, C.A., Gounaris, C.E.: A review of recent advanced in global optimization. J. Glob. Optim. 45, 3–38 (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chachuat, B., Singer, A.B., Barton, P.I.: Global methods for dynamic optimization and mixed-integer dynamic optimization. Ind. Eng. Chem. Res. 45, 8373–8392 (2006)CrossRefGoogle Scholar
  5. 5.
    Barton, P.I., Lee, C.K., Yunt, M.: Optimization of hybrid systems. Comput. Chem. Eng. 30, 1576–1589 (2006)CrossRefGoogle Scholar
  6. 6.
    Lin, Y.D., Stadtherr, M.A.: Deterministic global optimization of nonlinear dynamic systems. AIChE J. 53, 866–875 (2007)CrossRefGoogle Scholar
  7. 7.
    Lopez Cruz, I.L.: Efficient Evolutionary Algorithms for Optimal Control. Ph.D. thesis (2002)Google Scholar
  8. 8.
    Moiseev, N.N.: The Elements of the Optimal Systems Theory. Nauka, Moscow (1975). (in Russian)zbMATHGoogle Scholar
  9. 9.
    Bellman, R.: The Dynamical Programming. Princeton University Press, Princeton (1957)zbMATHGoogle Scholar
  10. 10.
    Krotov, V.F.: Global Methods in Optimal Control Theory. Marcel Dekker Inc., New York (1996)zbMATHGoogle Scholar
  11. 11.
    Strekalovsky, A.S., Yanulevich, M.V.: The global search in optimal control problem with cost terminal functional which is presented by difference of two convex functions. J. Comput. Math. Math. Phys. 48, 1187–1201 (2008). (in Russian)Google Scholar
  12. 12.
    Zarodnyuk, T.S., Gornov, A.Y.: A technique of finding global extremum in optimal control problems. Modern techniques. System analysis. Simulation 3, 70–76 (2008). (in Russian)Google Scholar
  13. 13.
    Gornov, A.Y., Zarodnyuk, T.S.: The “curvilinear search” method of global extremum in optimal control problems. Modern techniques. System analysis. Simulation 3, 19–26 (2009). (in Russian)Google Scholar
  14. 14.
    Krotov, V.F., Gurman, V.I.: The Methods and Problems of Optimal Control. Nauka, Moscow (1973). (in Russian)Google Scholar
  15. 15.
    Dykhta, V., Samsonyuk, O.: Some applications of Hamilton-Jacobi inequalities for classical and impulsive optimal control problems. Eur. J. Control. 17, 55–69 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dykhta, V.A., Sorokin, S.P.: Hamilton-Jacobi inequalities and the optimality conditions in the problems of control with common end constraints. Autom. Remote Control. 72, 1808–1821 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tolstonogov, A.A.: Differential Inclusions in a Banach Space, Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  18. 18.
    Shary, S.P.: Randomized algorithms in interval global optimization. Numer. Anal. Appl. 1(4), 376–389 (2008)CrossRefGoogle Scholar
  19. 19.
    Gamkrelidze, R.V.: Principles of Optimal Control Theory. Plenum Press, New York (1978)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mordukhovich, B.S.: Approximation Methods in Optimization and Control Problems. Nauka, Moscow (1988). (in Russian)zbMATHGoogle Scholar
  21. 21.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  22. 22.
    Gornov, A.Y., Zarodnyuk, T.S.: Tunneling algorithm for solving nonconvex optimal control problems. In: Chinchuluun, A., et al. (eds.) Optimization, Simulation, and Control. Springer Optimization and Its Applications, vol. 76, pp. 289–299. Springer, Berlin (2013)CrossRefGoogle Scholar
  23. 23.
    Zhigljavsky, A., Zilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)zbMATHGoogle Scholar
  24. 24.
    Gornov, A.Y., Zarodnyuk, T.S., Madzhara, T.I., Daneyeva, A.V., Veyalko, I.A.: A collection of test multiextremal optimal control problems. In: Chinchuluun, A., et al. (eds.) Optimization, Simulation, and Control. Springer Optimization and Its Applications, vol. 76, pp. 257–274. Springer, Berlin (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Alexander Yu. Gornov
    • 1
    Email author
  • Tatiana S. Zarodnyuk
    • 1
  • Anton S. Anikin
    • 1
  • Evgeniya A. Finkelstein
    • 1
  1. 1.Matrosov Institute for Systems Dynamics and Control Theory of SB RASIrkutskRussia

Personalised recommendations