Automation and Remote Control

, Volume 66, Issue 12, pp 1972–1982 | Cite as

Sufficient Conditions for Nesting the Reachable Sets of Two Smooth Control Systems of Constant Rank and Linear in Phase Variables

  • M. V. Topunov
Determinate Systems
  • 22 Downloads

Abstract

Sufficient conditions for nesting the reachable sets of smooth control systems of constant rank and linear in phase variables are formulated. Sets that are reachable from an arbitrary point in time T and in a time not greater than T are determined. Examples are given.

Keywords

Control System Mechanical Engineer System Theory Arbitrary Point Phase Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. 1.
    Vakhrameev, S.A., Relay Theorems and Related Topics, Trudy Mat. Inst. Ross. Akad. Nauk, 1998, vol. 220, pp. 49–112.MathSciNetMATHGoogle Scholar
  2. 2.
    Agrachev, A.A. and Sachkov, Yu.L., Geometricheskaya teoriya upravleniya (A Geometric Control Theory), Moscow: Fizmatlit, 2005.Google Scholar
  3. 3.
    Lee, E.B. and Markus, L., Foundations of Optimal Control Theory, New York: Wiley, 1973. Translated under the title Osnovy teorii optimal'nogo upravleniya, Moscow: Mir, 1973.Google Scholar
  4. 4.
    Chernous'ko, F.L., Otsenivanie fazovogo sostoyaniya sistem. Metod ellipsoidov (Phase State of Systems: An Ellipsoidal Estimation Method), Moscow: Nauka, 1988.Google Scholar
  5. 5.
    Brockett, R.W., On the Reachable Set for Bilinear Systems, Lect. Notes Econ. Math. Syst., 1975, vol. 111, pp. 54–63.MathSciNetMATHGoogle Scholar
  6. 6.
    Andreev, Yu.N., Differential Geometric Methods in Control Theory, Avtom. Telemekh., 1982, no. 10, pp. 5–46.Google Scholar
  7. 7.
    Bressan, A., Directional Convexity and Finite Optimality Conditions, J. Math. Anal. Appl., 1987, vol. 125, no.1, pp. 234–246.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Krener, A.J. and Schattler, H., The Structure of Small-Time Reachable Sets in Low Dimensions, SIAM J. Control. Optimiz., 1989, vol. 27, no.1, pp. 120–147.MathSciNetGoogle Scholar
  9. 9.
    Davydov, A.A., Properties of the Reachable Boundary in Two-Dimensional Control Systems, Usp. Mat. Nauk, 1982, vol. 37, no.3, pp. 183, 184.MathSciNetMATHGoogle Scholar
  10. 10.
    Formal'skii, A.M., Angular Points of the Boundaries of Reachable Domains, Prikl. Mat. Mekh., 1983, vol. 47, no.4, pp. 566–574.MathSciNetGoogle Scholar
  11. 11.
    Sergeev, S.I., Construction of Reachable Sets for a Class of Many-Step Control Processes, Avtom. Telemekh., 2002, no. 6, pp. 57–63.Google Scholar
  12. 12.
    Sirotin, A.N. and Formal'skii, A.M., Reachable Domains and Controllability of Linear Discrete Systems, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2002, no. 4, pp. 5–16.Google Scholar
  13. 13.
    Sirotin, A.N. and Formal'skii, A.M., Reachable and Controllability of Discrete Systems under Bounded Magnitude and Pulse Control Actions Avtom. Telemekh., 2003, no. 12, pp. 17–32.Google Scholar
  14. 14.
    Topunov, M.V., Convexity of Reachable Sets of Quasi-Commutative Bilinear Systems, Avtom. Telemekh., 2003, no. 8, pp. 44–53.Google Scholar
  15. 15.
    Topunov, M.V., Convexity of the Reachable Set of a Bilinear Control System, Prikl. Mat. Mekh., 2003, vol. 63, no.5, pp. 752–758.MathSciNetGoogle Scholar
  16. 16.
    Topunov, M.V., Convexity of the Reachable Set of a Smooth Linear Control System in Phase Variables, Avtom. Telemekh., 2004, no. 11, pp. 79–85.Google Scholar
  17. 17.
    Clarke, F.H., Optimization and Nonsmooth Analysis, New York: Wiley, 1983. Translated under the title Optimizatsiya i negladkii analiz, Moscow: Nauka, 1988.Google Scholar
  18. 18.
    Aubin, J.P. and Ekeland, I., Applied Nonlinear Analysis, New York: Wiley, 1984. Translated under the title Prikladnoi nelineinyi analiz, Moscow: Mir, 1988.Google Scholar
  19. 19.
    Vakhrameev, S.A., An Existence Theorem for Nonlinear Time-Optimal Problems, Diff. Uravn., 1999, vol. 35, no.4, pp. 565–567.MathSciNetMATHGoogle Scholar
  20. 20.
    Gantmakher, F.R., Teoriya matrits, Moscow: Nauka, 1967. Translated under the title The Theory of Matrices, New York: Chelsea, 1959.Google Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2005

Authors and Affiliations

  • M. V. Topunov
    • 1
  1. 1.Moscow State Institute of Steel and AlloysMoscowRussia

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