A maximum principle for smooth optimal impulsive control problems with multipoint state constraints

Article

Abstract

A nonlinear optimal impulsive control problem with trajectories of bounded variation subject to intermediate state constraints at a finite number on nonfixed instants of time is considered. Features of this problem are discussed from the viewpoint of the extension of the classical optimal control problem with the corresponding state constraints. A necessary optimality condition is formulated in the form of a smooth maximum principle; thorough comments are given, a short proof is presented, and examples are discussed.

Keywords

impulsive control trajectories of bounded variation intermediate state constraints maximum principle smooth constrained optimal control problems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. T. Zavalishchin and A. N. Sesekin, Impulsive Processes: Models and Applications (Nauka, Moscow, 1991) [in Russian].MATHGoogle Scholar
  2. 2.
    V. I. Gurman, Extension Principle in Control Problems, 2nd ed. (Nauka, Moscow, 1997) [in Russian].MATHGoogle Scholar
  3. 3.
    B. M. Miller and E. Ya. Rubinovich, Optimization of Dynamical Systems with Impulsive Controls (Nauka, Moscow, 2005) [in Russian].Google Scholar
  4. 4.
    V. A. Dykhta and O. N. Samsonyuk, Optimal Impulsive Control with Applications, 2nd ed. (Fizmatlit, Moscow, 2003) [in Russian].Google Scholar
  5. 5.
    L. T. Ashchepkov, “A General Maximum Principle for Systems with Intermediate Constraints on the Trajectory,” in Control and Optimization (Far East Division of the Russian Academy of Sciences, Vladivostok, 1991), pp. 16–27 [in Russian].Google Scholar
  6. 6.
    A. V. Dmitruk and A. M. Kaganovich, “A Maximum Principle for Optimal Control Problems with Intermediate Constraints,” in Nonlinear Dynamical Systems and Control (Nauka, Moscow, 2008), Vol. 6, pp. 1–40 [in Russian].Google Scholar
  7. 7.
    A. V. Arutyunov and A. I. Okoulevitch, “Necessary Optimality Conditions for Optimal Control Problems with Intermediate Constraints,” J. Dynamical and Control Systems 4(1), 49–58 (1998).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    F. H. Clarke and R. B. Vinter, “Optimal Multiprocesses,” SIAM J. Control Optimizat. 27, 1072–1091 (1989).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    F. H. Clarke and R. B. Vinter, “Applications of Optimal Multiprocesses,” SIAM J. Control Optimizat. 27, 1048–1071 (1989).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. E. Caines, F. H. Clarke, X. Liu, and R. B. Vinter, “A Maximum Principle for Hybrid Optimal Control Problems with Pathwise State Constraints,” in Proc. of the 45th IEEE Conference on Decision and Control, Phoenix, 2006, pp. 4821–4825.Google Scholar
  11. 11.
    M. Garavello and B. Piccoli, “Hybrid Necessary Principle,” SIAM J. Control Optimizat. 43, 1867–1887 (2005).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    H. J. Sussmann, “A Maximum Principle for Hybrid Optimal Control Problems,” in Proc. of the 38th IEEE Conference on Decision and Control, Phoenix, 1999.Google Scholar
  13. 13.
    A. V. Arutyunov, D. Yu. Karamzin, and F. L. Pereira, “A Nondegenerate Maximum Principle for the Impulse Control Problem with State Constraints,” SIAM J. Control Optimizat. 43, 1812–1843 (2005).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D. Yu. Karamzin, “Necessary Conditions of the Minimum in Impulsive Control Problems with Vector Measures,” J. Math. Sci. 139, 7087–7150 (2006).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    F. L. Pereira, “A Maximum Principle for Impulsive Control Problems with Statte Constraints,” Comput. Appl. Math. 19(2), 137–155 (2000).MATHMathSciNetGoogle Scholar
  16. 16.
    G. N. Silva, I. S. Litvinchev, M. Rojas-Medar, and A. J. V. Brandao, “State Constraints in Optimal Impulsive Controls,” Comput. Appl. Math 19(2), 179–206 (2000).MATHMathSciNetGoogle Scholar
  17. 17.
    M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction (Birkhäuser, Basel, 1993).MATHGoogle Scholar
  18. 18.
    V. A. Dykhta and O. N. Samsonyuk, “Maximum Principle in Nonsmooth Optimal Impulsive Control Problems with Multipoint State Constraints,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 19–32 (2001).Google Scholar
  19. 19.
    L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Interscience, New York, 1962).Google Scholar
  20. 20.
    R. B. Vinter and F. L. Pereira, “A Maximum Principle for Optimal Processes with Discontinuous Trajectories,” SIAM J. Control Optimizat. 26, 205–229 (1988).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    G. N. Silva and R. B. Vinter, “Necessary Optimality Conditions for Optimal Impulsive Control Problem,” SIAM J. Control Optimizat. 35, 1829–1846 (1997).MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    V. A. Dykhta, “Necessary Optimality Conditions for Impulsive Processes under Constraints on the Image of the Control Measure,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 12, 1–9 (1996).Google Scholar
  23. 23.
    M. Motta and F. Rampazzo, “Space-Time Trajectories of Nonlinear Systems Driven by Ordinary and Impulsive Controls,” Differential Integral Equat. 8, 269–288 (1995).MATHMathSciNetGoogle Scholar
  24. 24.
    J. R. Dorroh and G. Ferreyra, “A Multistate, Multicontrol Problem with Unbounded Controls,” SIAM J. Control Optimizat. 32, 1322–1331 (1994).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Institute of System Dynamics and Control TheorySiberian Branch of the Russian Academy of SciencesIrkutskRussia

Personalised recommendations