Perspectives in Control Theory pp 129-142 | Cite as

# Fuller’s Phenomena

Chapter

## Abstract

In his paper [F], Fuller studied the following optimal control problem. In the state space ℝ^{2} 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 ℝ^{2}, he wants to determine the trajectories (*x̂, ŷ, û*) :[0, *T̂*] → ℝ^{2} × [-1, +1] of the system, starting at *a*, ending at 0 and minimizing the cost \(
\frac{1}{2}\int_0^{\hat T} {\hat y{{(t)}^2}dt}
\).

## Keywords

Optimal Control Problem Open Neighborhood Poisson Bracket Optimal Trajectory Remainder Term
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

- [F]A.T. FULLER: “Study of an optimum non linear system” J. Electronic Control 15 (1963) pp 63–71.CrossRefGoogle Scholar
- [G]R. GAMKRELIDZE: “Principles of optimal control theory”, Plenum (1978).CrossRefGoogle Scholar
- [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
- [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
- [R]E.P. RYAN: “Optimal relay and saturating system synthesis” I.E.E. Control Engineering Series no. 14, Peter Peregrinus (1982).Google Scholar
- [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

## Copyright information

© Springer Science+Business Media New York 1990