Fuller’s Phenomena

  • I. A. K. Kupka
Part of the Progress in Systems and Control Theory book series (PSCT, volume 2)


In his paper [F], Fuller studied the following optimal control problem. In the state space ℝ2 with coordinates (x, y) he considers the control system: \(\frac{{dx}}{{dt}} = u\quad \frac{{dy}}{{dt}} = x \) where the control u is restricted to the segment [-1, +1]. Given a point a in ℝ2, he wants to determine the trajectories (x̂, ŷ, û) :[0, ] → ℝ2 × [-1, +1] of the system, starting at a, ending at 0 and minimizing the cost \( \frac{1}{2}\int_0^{\hat T} {\hat y{{(t)}^2}dt} \).


Optimal Control Problem Open Neighborhood Poisson Bracket Optimal Trajectory Remainder Term 


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  1. [F]
    A.T. FULLER: “Study of an optimum non linear system” J. Electronic Control 15 (1963) pp 63–71.CrossRefGoogle Scholar
  2. [G]
    R. GAMKRELIDZE: “Principles of optimal control theory”, Plenum (1978).CrossRefGoogle Scholar
  3. [K1]
    I.A.K. KUPKA: “Geometric theory of extremals in optimal control problems: I. The fold and maxwell case.” T.A.M.S. vol. 299 no 1 (Jan 1987) pp 225–243.Google Scholar
  4. [K2]
    I.A.K. KUPKA: “The ubiquity of Fuller’s phenomena” to appear in the proceedings of the workshop on Optimal Control at Rutgers University. H. Sussmann editor, M. Dekker publisher.Google Scholar
  5. [R]
    E.P. RYAN: “Optimal relay and saturating system synthesis” I.E.E. Control Engineering Series no. 14, Peter Peregrinus (1982).Google Scholar
  6. [S]
    H.J. SUSSMANN: in “Differential geometry control theory” R.W. Brockett, R.S. Millman, H.J. Sussmann ed., Birkhäuser PM no. 27 (1983).Google Scholar

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© Springer Science+Business Media New York 1990

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  • I. A. K. Kupka

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