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Avoidance Control on Time Scales

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Abstract

We consider dynamic systems on time scales under the control of two agents. One of the agents desires to keep the state of the system out of a given set regardless of the other agent’s actions. Leitmann’s avoidance conditions are proved to be valid for dynamic systems evolving on an arbitrary time scale.

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References

  1. Aggarwal, R., Leitmann, G.: Avoidance control. Trans. ASME Ser. G, J. Dyn. Syst. Meas. Control 94, 152–154 (1972)

    MathSciNet  Google Scholar 

  2. Barmish, B.R., Schmitendorf, W.E., Leitmann, G.: A note on avoidance control. Trans. ASME Ser. G, J. Dyn. Syst. Meas. Control 103(1), 69–70 (1981)

    Article  MathSciNet  Google Scholar 

  3. Corless, M., Leitmann, G.: Adaptive controllers for avoidance or evasion in an uncertain environment: some examples. Comput. Math. Appl. 18(1–3), 161–170 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. Corless, M., Leitmann, G., Skowronski, J.M.: Adaptive control for avoidance or evasion in an uncertain environment. Comput. Math. Appl. 13(1–3), 1–11 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Leitmann, G.: Guaranteed avoidance strategies. J. Optim. Theory Appl. 32(4), 569–576 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  6. Leitmann, G., Skowroński, J.: Avoidance control. J. Optim. Theory Appl. 23(4), 581–591 (1977)

    Article  MATH  Google Scholar 

  7. Leitmann, G., Skowroński, J.: A note on avoidance control. Optim. Control Appl. Methods 4(4), 335–342 (1983)

    Article  MATH  Google Scholar 

  8. Stipanović, D.M.: A Survey and some new results in avoidance control. In: Rodellar, J., Reithmeier, E. (eds.) Proceedings of the 15th International Workshop on Dynamics and Control, IWDC 2009, pp. 166–173

  9. Agarwal, R.P., Bohner, M.: Basic calculus on time scales and some of its applications. Results Math. 35(1–2), 3–22 (1999)

    MATH  MathSciNet  Google Scholar 

  10. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser, Boston (2001)

    MATH  Google Scholar 

  11. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)

    MATH  Google Scholar 

  12. Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control. American Institute of Mathematical Sciences (AIMS), Springfield (2007)

    MATH  Google Scholar 

  13. Cabada, A., Vivero, D.R.: Criterions for absolute continuity on time scales. J. Differ. Equ. Appl. 11(11), 1013–1028 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic, Dordrecht (1988). Translated from the Russian

    Google Scholar 

  15. Bartosiewicz, Z., Pawłuszewicz, E.: Realizations of linear control systems on time scales. Control Cybern. 35(4), 769–786 (2006)

    MATH  Google Scholar 

  16. DaCunha, J.J.: Lyapunov stability and Floquet theory for nonautonomous linear dynamic systems on time scales. Ph.D. thesis, Baylor University (2004)

  17. DaCunha, J.J.: Stability for time varying linear dynamic systems on time scales. J. Comput. Appl. Math. 176(2), 381–410 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Bartosiewicz, Z., Piotrowska, E., Wyrwas, M.: Stability, stabilization and observers of linear control systems on time scales. In: Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, USA (2007)

  19. Cabada, A., Vivero, D.R.: Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral: application to the calculus of Δ-antiderivatives. Math. Comput. Model. 43(1–2), 194–207 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Deniz, A.: Measure theory on time scales. MSc thesis, Graduate School of Engineering and Sciences of Izmir Institute of Technology, Turkey (2007)

  21. Kolmogorov, A.N., Fomīn, S.V.: Introductory Real Analysis. Dover, New York (1975). Translated from the second Russian edition and edited by Richard A. Silverman

    Google Scholar 

  22. Sontag, E.D.: Mathematical Control Theory. Springer, New York (1990)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    E. Pawłuszewicz & D. F. M. Torres

Authors
  1. E. Pawłuszewicz
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  2. D. F. M. Torres
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Correspondence to D. F. M. Torres.

Additional information

E. Pawłuszewicz on leave from Białystok Technical University, Poland. e-mail: epaw@pb.edu.pl.

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Pawłuszewicz, E., Torres, D.F.M. Avoidance Control on Time Scales. J Optim Theory Appl 145, 527–542 (2010). https://doi.org/10.1007/s10957-010-9694-1

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