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Invariance for impulsive control systems

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Abstract

The conventional concepts of invariance are extended in this article to include impulsive control systems represented by measure driven differential inclusions. Invariance conditions and some of their main features are derived. The solution concept plays a critical role in the extension of the conditions for conventional problems to the impulsive control context.

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References

  1. 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 

  2. 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 

  3. Clark, C., Clarke, F., and Munro, G., The Optimal Exploitation of Renewable Stocks, Econometrica, 1979, vol. 47, pp. 25–47.

    Article  MATH  Google Scholar 

  4. Baumeister, J., On Optimal Control of a Fishery, in NOLCOS’01—5th IFAC Symposium on Nonlinear Control Systems, 2001.

  5. Gurman, V.I., Modeli upravleniya prirodnymi resursami (Models of Natural Resources Control), Moscow: Nauka, 1981.

    Google Scholar 

  6. Brogliato, B., Nonsmooth Impact Mechanics: Models, Dynamics and Control, Lect. Notes Control Inform. Sci., Berlin: Springer-Verlag, 1996, vol. 220.

    Google Scholar 

  7. Marec, J.P., Optimal Space Trajectories, New York: Elsevier, 1979.

    MATH  Google Scholar 

  8. Arutyunov, A., Jacimovic, V., and Lobo Pereira F., Second Order Necessary Conditions for Optimal Impulsive Control Problems, European Control Conference, 2001.

  9. Bressan, A. and Rampazzo, F., Impulsive Control Systems with Commutative Vector Fields, J. Optim. Theory Appl., 1991, vol. 71, no. 1, pp. 67–83.

    Article  MATH  MathSciNet  Google Scholar 

  10. Bressan, A. and Rampazzo, F., Impulsive Control Systems without Commutativity Assumptions, J. Optim. Theory Appl., 1994, vol. 81, no. 3, pp. 435–457.

    Article  MATH  MathSciNet  Google Scholar 

  11. Dykhta, V., Impulse-trajectory Extension of Degenerate Optimal Control Problems, IMACS Ann. Comput. Appl. Math., 1990, vol. 8, pp. 103–109.

    MathSciNet  Google Scholar 

  12. Dykhta, V.A. and Samsonyuk, O.N., Optimal Impulse Control with Applications, Moscow: Nauka, 2000.

    Google Scholar 

  13. Gurman, V.I., On Optimal Processes with Unbounded Derivatives, Autom. Remote Control, 1972, vol. 17, pp. 14–21.

    Google Scholar 

  14. Kolokolnikova, G.A., Discontinuous Trajectories Optimality in the Nonlinear Optimal Control Problems, Proc. 13th World Congress of the IFAC, 1996, pp. 353–357.

  15. Miller, B.M., The Generalized Solutions of Nonlinear Optimization Problems with Impulse Control, SIAM J. Control Optim., 1996, vol. 34.

  16. Motta, M. and Rampazzo, F., Space-time Trajectories of Nonlinear Systems Driven by Ordinary and Impulse Controls, Diff. Integral Equat., 1995.

  17. Rishel, R., An Extended Pontryagin Principle for Control Systems whose Control Laws Contain Measures, SIAM J. Control, 1965, vol. 3, pp. 191–205.

    MathSciNet  Google Scholar 

  18. Vinter, R.B. and Pereira, F.M., A Maximum Principle for Optimal Processes with Discontinuous Trajectories, SIAM J. Control Optim., 1988, vol. 26, pp. 205–229.

    Article  MATH  MathSciNet  Google Scholar 

  19. Silva, G.N. and Vinter, R.B., Necessary Conditions for Optimal Impulsive Control Problems, SIAM J. Control Optim., 1997, vol. 35, pp, 1829–1846.

    Article  MATH  MathSciNet  Google Scholar 

  20. Dykhta, V. and Derenko, N., Numerical Methods based on Generalized Stationary Condition for Impulsive Control Problems, in Proc. Workshop Comput. Logics, Algebra Intelligence Control, Irkutsk, 1994, vol. 2, pp. 59–70.

    Google Scholar 

  21. Baturin, V. and Goncharova, E., Improvement Method based on Approximate Representation of Reachable Set, Autom. Remote Control, 1999, no. 11, pp. 19–29.

  22. Baturin, V. and Ourbanovich, D., Approximate Optimal Control Methods based on Extension Principle, Novosibirsk: Nauka, 1997.

    Google Scholar 

  23. Gurman, V.I., Baturin, V.A., Moskalenko, A.I., et al., Methods of Improvement in Computational Experiments, Novosibirsk: Nauka, 1988.

    Google Scholar 

  24. Pereira, F.L. and Silva, G.N., Stability for Impulsive Control Systems, Dynam. Syst., 2002, vol. 17, no. 4, pp. 421–434.

    Article  MATH  MathSciNet  Google Scholar 

  25. Pereira, F.L. and Silva, G.N., Lyapunov Stability of Measure Driven Differential Inclusions, Diff. Uravn., 2004, vol. 40, no. 8, pp. 1059–1067.

    MathSciNet  Google Scholar 

  26. Aubin, J.-P., Impulse Differential Equations and Hybrid Systems: A Viability Approach, Berkeley: Univ. of California, 2000.

    Google Scholar 

  27. Branicky, M., Borkar, V.S., and Mitter, S.K., A Unified Framework for Hybrid Control: Model and Optimal Control Theory, IEEE Trans. Automat. Control, 1998, vol. 43, no. 1, pp. 31–45.

    Article  MATH  MathSciNet  Google Scholar 

  28. Clarke, F.H., Ledyaev, Yu., Stern, R., and Wolenski, P., Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, Springer-Verlag, 1998, vol. 178.

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Authors and Affiliations

  1. University of Porto, Porto, Portugal

    F. Lobo Pereira

  2. University of British Columbia, Vancouver, Canada

    G. Nunes Silva

  3. University of Campinas, Campinas, Brasil

    V. Oliveira

Authors
  1. F. Lobo Pereira
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  2. G. Nunes Silva
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  3. V. Oliveira
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Additional information

Original Russian Text © F. Lobo Pereira, G. Nunes Silva, V. Oliveira, 2008, published in Avtomatika i Telemekhanika, 2008, No. 5, pp. 57–71.

This work has been presented on the Symposium “Generalized Solution in Control Problems.”

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Pereira, F.L., Silva, G.N. & Oliveira, V. Invariance for impulsive control systems. Autom Remote Control 69, 788–800 (2008). https://doi.org/10.1134/S0005117908050068

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