Skip to main content
SpringerLink
Log in

Invariant sets for the nonlinear impulsive control systems

  • Nonlinear Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

The nonlinear impulsive control system with trajectories of bounded variation is considered. The notions of strong and weak V-invariance of a closed set relative to the impulsive control system are introduced. These notions are adapted to the system nonautonomy in the so-called “fast” time, in which the impulsive dynamics occurs. Invariance criteria in the form of the proximal Hamilton-Jacobi equalities were proved.

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. Aubin, J.-P. and Cellina, A., Differential Inclusions, Berlin: Springer, 1984.

    Book  MATH  Google Scholar 

  2. Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., et al., Nonsmooth Analysis and Control Theory, New York: Springer, Grad. Texts in Math, vol. 178, 1998.

  3. Guseinov, Kh.G. and Ushakov, V.N., Strongly and Weakly Invariant Sets Relative to the Differential Inclusion, Their Derivatives, and Application to the Problems Control, Diff. Uravn., 1990, vol. 26, no. 11, pp. 1888–1894.

    MathSciNet  Google Scholar 

  4. Pereira, F.L., Silva, G.N., and Oliveira, V., Invariance for Impulsive Control Systems, Autom. Remote Control, 2008, vol. 69, no. 5, pp. 788–800.

    Article  MATH  MathSciNet  Google Scholar 

  5. Samsonyuk, O.N., Strong and Weak Invariance for Nonlinear Impulsive Control Systems, in Book of Abstracts of IFAC WC 2011, 2011, pp. 3480–3485.

    Google Scholar 

  6. Gurman, V.I., Printsip rasshireniya v zadachakh upravleniya (Principle of Extension in the Problems of Control), Moscow: Nauka, 1997, 2nd ed.

    MATH  Google Scholar 

  7. Dykhta, V.A. and Samsonyuk, O.N., Optimal’noe impul’snoe upravlenie s prilozheniyami (Optimal Impulsive Control with Applications), Moscow: Fizmatlit, 2003, 2nd ed.

    MATH  Google Scholar 

  8. Zavalishchin, S.T. and Sesekin, A.N., Impul’snye protsessy: modeli i prilozheniya (Impulsive Processes: Models and Applications), Moscow: Nauka, 1991.

    MATH  Google Scholar 

  9. Miller, B.M., Optimality Condition in the Control Problem for a System Described by a Measure Differential Equation, Autom. Remote Control, 1982, vol. 43, no. 6, part 1, pp. 752–761.

    MATH  Google Scholar 

  10. Miller, B.M., Conditions for the Optimality in Problems of Generalized Control. I, II, Autom. Remote Control, 1992, vol. 53, no. 3, part 1, pp. 362–370; no. 4, part 1, pp. 505–513.

    Google Scholar 

  11. Miller, B.M., Method of Discontinuous Time Change in Problems of Control of Impulse and Discrete-Continuous Systems, Autom. Remote Control, 1993, vol. 54, no. 12, part 1, pp. 1727–1750.

    Google Scholar 

  12. Miller, B.M., The Generalized Solutions of Nonlinear Optimization Problems with Impulse Control, SIAM J. Control Optim., 1996, vol. 34, pp. 1420–1440.

    Article  MATH  MathSciNet  Google Scholar 

  13. Miller, B.M. and Rubinovich, E.Ya., Optimizatsiya dinamicheskikh sistem s impul’snymi upravleniyami (Optimization of the Dynamic Systems with Impulsive Controls), Moscow: Nauka, 2005.

    Google Scholar 

  14. Miller, B.M. and Rubinovich, E.Ya., Impulsive Control with Impulsive Actions of Two Types, Autom. Remote Control, 2009, vol. 70, no. 11, pp. 1795–1813.

    Article  MATH  MathSciNet  Google Scholar 

  15. Sesekin, A.N., On Continuous Dependence on the Right Sides and Stability of the Approximated Solutions of Differential Equations with Products of Discontinuous Functions by Generalized Ones, Differ. Uravn., 1986, no. 11, pp. 2009–2011.

    Google Scholar 

  16. Sesekin, A.N., On Sets of Discontinuous Solutions of the Nonlinear Differential Equations, Izv. Vyssh. Uchebn. Zaved., Mat., 1994, no. 6, pp. 83–89.

    Google Scholar 

  17. Sesekin, A.N., Properties of the Attainability Set of a Dynamic System with Impulsive Control, Autom. Remote Control, 1994, vol. 55, no. 2, part 1, pp. 190–195.

    MATH  MathSciNet  Google Scholar 

  18. Sesekin, A.N., Dynamic Systems with Nonlinear Impulsive Structure, Tr. Inst. Mat. Mekh. UrO RAN, 2000, no. 6, pp. 497–510.

    Google Scholar 

  19. Sesekin, A.N. and Fetisova, Yu.V., Functional-Differential Equations in the Space of Bounded-variation Functions, Tr. Inst. Mat. Mekh. UrO RAN, 2009, vol. 15, no. 4, pp. 227–233.

    MathSciNet  Google Scholar 

  20. Arutyunov, A.V., Karamzin, D.Yu., and Pereira, F.L., A Nondegenerate Maximum Principle for the Impulse Control Problem with State Constraints, SIAM J. Control Optim., 2005, vol. 43, no. 5, pp. 1812–1843.

    Article  MATH  MathSciNet  Google Scholar 

  21. Arutyunov, A.V., Karamzin, D.Yu., and Pereira, F.L., On Constrained Impulsive Control Problems, J. Math. Sci., 2010, vol. 165, no. 6, pp. 654–688.

    Article  MATH  MathSciNet  Google Scholar 

  22. Motta, M. and Rampazzo, F., Space-Time Trajectories of Nonlinear Systems Driven by Ordinary and Impulsive Controls, Differ. Integr. Equat., 1995, vol. 8, pp. 269–288.

    MATH  MathSciNet  Google Scholar 

  23. Motta, M. and Rampazzo, F., Nonlinear Systems with Unbounded Controls and State Constraints: A Problem of Proper Extension, NoDEA, 1996, vol. 3, pp. 191–216.

    Article  MATH  MathSciNet  Google Scholar 

  24. Pereira, F.L. and Silva, G.N., Necessary Conditions of Optimality for Vector-Valued Impulsive Control Problems, Syst. Control Lett., 2000, vol. 40, pp. 205–215.

    Article  MATH  MathSciNet  Google Scholar 

  25. Silva, G.N. and Vinter, R.B., Measure Differential Inclusions, J. Math. Anal. Appl., 1996, vol. 202, pp. 727–746.

    Article  MATH  MathSciNet  Google Scholar 

  26. Vinter, R.B., Optimal Control, Boston: Birkhauser, 2000.

    MATH  Google Scholar 

  27. Dykhta, V.A. and Samsonyuk, O.N., Hamilton-Jacoby Inequalities in the Problems of Control of Impulsive Dynamic Systems, in Trudy Mat. Inst. Steklov. RAN, 2010, vol. 271, pp. 93–110.

    MathSciNet  Google Scholar 

  28. Dykhta, V. and Samsonyuk, O., Some Applications of Hamilton-Jacobi Inequalities for Classical and Impulsive Optimal Control Problems, Eur. J. Control, 2011, vol. 17, pp. 55–69.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences, Irkutsk, Russia

    O. N. Samsonyuk

Authors
  1. O. N. Samsonyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. N. Samsonyuk.

Additional information

Original Russian Text © O.N. Samsonyuk, 2015, published in Avtomatika i Telemekhanika, 2015, No. 3, pp. 44–61.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Samsonyuk, O.N. Invariant sets for the nonlinear impulsive control systems. Autom Remote Control 76, 405–418 (2015). https://doi.org/10.1134/S0005117915030054

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117915030054

Keywords

Search

Navigation