Skip to main content
Log in

Fuller’s phenomenon: Review

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. V. I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989).

    MATH  Google Scholar 

  2. D. J. Bell and M. Boissard, “Necessary conditions at the junction of singular and non-singular control subarcs,” Int. J. Contr., 29, No. 6, 987–990 (1979).

    MathSciNet  Google Scholar 

  3. Ya. M. Berschanskii, “Chattering solutions to discontinuous differential equations,” in: Problems of Control of Multiconnected Systems [in Russian], Nauka, Moscow (1983), pp. 29–42.

    Google Scholar 

  4. Ya. M. Berschanskii, “Behavior of optimal trajectories in a vicinity of the terminal set,” in: Research on the Theory of Multiconnected Systems [in Russian], VNIISI, Moscow (1982), pp. 76–89.

    Google Scholar 

  5. Ya. M. Berschanskii, “Trajectories of linear systems with a relay-type nonlinearity,” Avtomat. Telenekh., No. 7, 19–27 (1982).

    Google Scholar 

  6. Ya. M. Berschanskii, “Some singularities of synthesis for nonlinear systems with a degenerate quadratic performance index,” in: Research in and Optimization of Multiconnected Systems [in Russian], Nauka, Moscow, pp. 36–58 (1979).

    Google Scholar 

  7. Ya. M. Berschanskii, “Analytical investigation of optimal trajectories at points of conjugation with a singular arc,” in: Research in and Optimization of Multiconnected Systems [in Russian], Nauka, Moscow, pp. 59–70 (1979).

    Google Scholar 

  8. Ya. M. Berschanskii, “Conjugation of singular and nonsingular subarcs of optimal control,” Avtomat. Telemekh., No. 3, 5–11 (1979).

    Google Scholar 

  9. Ya. M. Berschanskii, “Some problems of synthesis for linear systems with a quadratic performance index,” Avtomat. Telemekh., No. 3, 5–14 (1976).

    Google Scholar 

  10. V. F. Borisov, “Construction of optimal synthesis with a chattering mode,” Sov. Math. Dokl., 38, No. 2, 332–336 (1989).

    MATH  Google Scholar 

  11. V. F. Borisov, “On the number of limit cycles in the factor system of the n-dimensional Fuller problem,” Mat. Sb., 187, No. 12, 3–20 (1996).

    MathSciNet  Google Scholar 

  12. V. F. Borisov, “On the two-dimensional submanifolds of a class of discontinuous Hamiltonian systems,” Fund. Prikl. Mat. (in press).

  13. M. Z. Borshchevskii and I. V. Ioslovich, “The problem of the optimum rapid braking of an axisymmetric solid rotating around its center of mass,” Prikl. Mat. Mekh., 49, No. 1, 35–42 (1985); English transl.: Appl. Math. Mech., 49, No. 1, 24–30 (1985).

    Google Scholar 

  14. J. V. Breakwell and J. F. Dixon, “Minimum-fuel rocket trajectories involving intermediate thrust arcs,” J. Optimiz. Theory Appl., 17, No. 5, 465–479 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Brunovsky and J. Mallet-Paret, “Switchings of optimal controls and the equation y (4)+y αsgn y=0, 0<α<1,” Čas. Pěst. Mat., 110, No. 3, 302–313 (1985).

    MATH  MathSciNet  Google Scholar 

  16. S. V. Chukanov and A. A. Milyutin, “Qualitative study of singularities for extremals of the quadratic optimal control problem.” Russian J. Math. Phys., 2, No. 1, 31–48 (1994).

    MATH  MathSciNet  Google Scholar 

  17. V. V. Dikusar and A. A. Milyutin, Qualitative and Numerical Methods in the Maximum Principle [in Russian], Nauka, Moscow (1989).

    MATH  Google Scholar 

  18. C. M. Dorling and E. P. Ryan, “Minimization of nonquadratic cost functionals for a third order saturating system,” Int. J. Control, 34, No. 2, 231–258 (1981).

    MATH  MathSciNet  Google Scholar 

  19. V. R. Telesnin, “On a problem of optimizing transition processes,” Proc. Math. Inst. Akad. Nauk SSSR, 166, No. 1, 261–271 (1986).

    MATH  Google Scholar 

  20. A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides [in Russian], Nauka, Moscow (1985); English transl.: Reidel (1989).

    Google Scholar 

  21. A. T. Fuller, “Relay control systems optimized for various performance criteria,” in: Proc. First World Congress IFAC, Moscow, 1960, Butterworths, London (1961), pp. 510–519.

    Google Scholar 

  22. A. T. Fuller, “Further study of an optimum non-linear control system,” J. Electr. Control, 17, No. 3, 283–301 (1964).

    MathSciNet  Google Scholar 

  23. A. T. Fuller, “Absolute optimality of a non-linear control system with integral-square error criterion,” J. Electr. Control, 17, No. 3, 301–317 (1964).

    MathSciNet  Google Scholar 

  24. A. T. Fuller and P. E. Grensted, “Minimization of integral square error for nonlinear control systems of third and higher order,” Int. J. Control, 2, No. 1, 33–73 (1965).

    Article  Google Scholar 

  25. P. Hartman, Ordinary Differential Equations, J. Wiley & Sons, New York-London-Sydney (1964).

    MATH  Google Scholar 

  26. M. W. Hirsh, C. C. Pugh, and M. Shub, Invariant Manifolds, Springer-Verlag, Berlin (1977).

    Google Scholar 

  27. D. E. Johansen, “Solutions of linear-mean square estimation problem when process statistics are undefined,” in: Joint Automatic Control Conf., Troy, New York (1965), pp. 64–75.

  28. H. J. Kelley, R. E. Kopp, and H. G. Moyer, “Singular extremals,” in Topics in Optimization (ed. by G. Leitmann), Academic Press, New York (1967), pp. 63–103.

    Google Scholar 

  29. I. Kupka, “The ubiquity of Fuller’s phenomenon,” in: Nonlinear Controllability and Optimal Control, Monograph Textbook, Pure Appl. Math., No. 133 (ed. by H. Sussman), Dekker, New York (1990), pp. 313–350.

    Google Scholar 

  30. I. Kupka, “Fuller’s phenomena,” in: Perspectives in Control Theory (Sielpia, 1988), Progr. Systems Control Theory, 2, Birkhäuser, Boston (1990), pp. 129–142.

    Google Scholar 

  31. I. Kupka, “Geometric theory of extremals in optimal control problems: The fold and Maxwell case,” Trans. Amer. Math. Soc., 299, No. 1, 225–243 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  32. I. Kupka, “Geometric theory of extremals. Fuller Phenomenon,” in: Proc XXIV Conf. Decision and Control, Ft. Lauderdale, Florida (1985), pp. 711–713.

  33. R. M. Lewis, “Definition of order and junction condition in singular control problems,” SIAM J. Control Optim., 18, No. 1, 21–32 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  34. G. G. Magaril-Il’yaev, “Kolmogorov inequalities on the half-line,” Vestn. MGU, Mat., Mekh. No. 5, 31–41 (1976); English transl.: Moscow Univ. Math. Bull., 31, No. 5-6, 25–32 (1976).

    Google Scholar 

  35. L. A. Manita, “The behavior of extremals in the neighborhood of singular regimes and nonsmooth Lyapunov functions in optimal control problems,” Fund. Prikl. Mat., 2, No. 2, 411–447 (1996).

    MATH  MathSciNet  Google Scholar 

  36. C. Marchal, “Chattering arcs and chattering controls,” J. Optimiz. Theory Appl., 11, No. 5, 441–446 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  37. J. P. McDanell and W. F. Powers, “Necessary conditions for joining optimal and non-singular subarcs,” SIAM J. Contr. Optim., 9, No. 2, 161–172 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  38. M. V. Meerov, A. V. Akhetzyanov, Ya. M. Berschanskii, et al., Multiconnected Control Systems [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  39. A. A. Milyutin, “On the equation ψ (4)=−sign ψ,” Tr. VNIISI, No. 1, 76–84 (1990).

    Google Scholar 

  40. A. A. Milyutin, “An example of an optimal control problem whose extremals possess a continual set of discontinuities of the control fuction,” Russian J. Math. Phys., 1, No. 3, 397–402 (1993).

    MATH  MathSciNet  Google Scholar 

  41. A. A. Milyutin, A. E. Ilyutovich, N. P. Osmolovskiî, and S. V. Chukanov, Optimal Control in Linear Systems [in Russian], Nauka, Moscow (1993).

    MATH  Google Scholar 

  42. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, 3rd ed. [in Russian], Nauka, Moscow (1976); English transl. of 2nd ed.: Wiley, 1962 and MacMillan, 1964.

    Google Scholar 

  43. H. M. Robbins, “Junction phenomena for optimal control with state-variable inequality constraints of third order,” J. Optim. Theory Appl., 31, No. 1, 85–99 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  44. H. M. Robbins, “A generalized Legendre-Clebsh condition for singular cases of optimal control,” IBM Res. Dev., 11 (1967).

  45. E. P. Ryan, “Time-optimal feedback control laws for certain third-order relay control systems,” Int. J. Contr., 20, No. 6, 881–913 (1974).

    Article  MATH  Google Scholar 

  46. E. P. Ryan, “Singular optimal controls for second order saturating systems,” Int. J. Contr., 30, No. 4, 549–564 (1979).

    Article  MATH  Google Scholar 

  47. D. B. Silin, “Linear time-optimal problems with control discontinuities at a set of positive measure,” Izv. Akad Nauk SSSR, Ser. Mat., 48, No. 4, 754–764 (1984).

    MathSciNet  Google Scholar 

  48. I. O. Vashkov, “Time-optimal problems to the infinity,” Vestn. MGU, Ser. 1, Mat., Mekh., No. 1, 36–40 (1990).

    MathSciNet  Google Scholar 

  49. L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders Co., Philadelphia-London-Toronto (1969).

    MATH  Google Scholar 

  50. M. I. Zelikin, Optimal control of rigid body rotation, Dokl. Ross. Akad. Nauk, 346, No. 3 (1996).

  51. M. I. Zelikin, “The Fuller phenomenon in problems of vibration of two linked oscillators,” J. Math. Sci., 78, No. 5, 626–631 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  52. M. I. Zelikin, “One-parameter families of solutions to a class of PDE optimal control problems,” Contemp. Math., 209, 339–349 (1997).

    MathSciNet  Google Scholar 

  53. M. I. Zelikin, “Irregularity of optimal control in the regular extremal problems,” Fund. Prikl. Mat., 1, No. 2, 399–408 (1995).

    MATH  MathSciNet  Google Scholar 

  54. M. I. Zelikin and V. F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering, Birkhäuser, Boston (1994).

    MATH  Google Scholar 

  55. M. I. Zelikin and V. F. Borisov, “Regimes with increasingly more frequent switchings in optimal problems,” Tr. Mat. Inst. Akad. Nauk SSSR, 197, 85–167 (1991); English transl.: Proc. of Steklov Inst. Math., No. 1, 95–186 (1993).

    MATH  Google Scholar 

  56. M. I. Zelikin and V. F. Borisov, “Fields of optimal trajectories containing singular second order extremals and extremals with frequent swithchings,” Dokl. Akad. Nauk SSSR, 304, No. 5, 1050–1053 (1989); English transl.: Sov. Math. Dokl., 39, No. 1, 188–191 (1989).

    MathSciNet  Google Scholar 

  57. M. I. Zelikin and V. F. Borisov, “Synthesis in problems of optimal control containing a trajectory with switchings whose frequency increases and second-order singular trajectories,” Math. Notes Acad. Sci. USSR, 47, No. 1-2, 41–49 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  58. M. I. Zelikin and V. F. Borisov, “Modes with switchings of increasing frequency in the problem of controlling a robot,” Prikl. Mat. Mekh., 52, No. 6, 934–946 (1988); English transl.: Appl. Math. Mech. 52, No. 6, 731–738 (1988).

    MathSciNet  Google Scholar 

  59. M. I. Zelikin and V. F. Borisov, “Chattering in Lawden’s problem of space navigation,” in: Inter. Aerospace Congress. Theory, Applications, Technologies. Abstracts, Moscow (1994), p. 448.

Download references

Authors

Additional information

This work was supported by the Russian Foundation for Basic Research, project No. 98-01-00535.

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya, Tematicheskie Obzory. Vol. 59, Dinamicheskie Sistemy-8, 1998.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borisov, V.F. Fuller’s phenomenon: Review. J Math Sci 100, 2311–2354 (2000). https://doi.org/10.1007/s10958-000-0001-9

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-000-0001-9

Keywords

Navigation