The brachistochrone for a steerable particle moving on a 1D curved surface in a gravity field is solved using an optimal control formulation with state feedback. The process begins with a derivation of a fourth-order open-loop plant model with the system input being the body yaw rate. Solving for the minimum-time control law entails introducing four costates and solving the Euler–Lagrange equations, with the Hamiltonian being stationary with respect to the control. Also, since the system is autonomous, the Hamiltonian must be zero. A two-point boundary value problem results with a transversality condition, and its solution requires iteration of the initial bearing angle so the integrated trajectory runs through the final point. For this choice of control, the Legendre–Clebsch necessary condition is not satisfied. However, the $k=1$ generalized Legendre–Clebsch necessary condition from singular control theory is satisfied for all numerical simulations performed, and optimality is assured. Simulations in MATLAB® exercise the theory developed and illustrate application such as to ski racing and minimizing travel time over either a concave or undulating surface when starting from rest. Lastly, a control law singularity in particle speed is overcome numerically.

1.
Bernoulli
,
J.
, 1697, “
Jacobi Bernoulli solutio problematum fraternorum
,”
Acta Eruditorum
, p.
214
.
2.
Lipp
,
S. C.
, 1997, “
Brachistochrone With Coulomb Friction
,”
SIAM J. Control Optim.
0363-0129,
35
(
2
), pp.
562
584
.
3.
Euler
,
L.
, 1744, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Guadentes sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti, Lausanne, Geneva.
4.
Lind
,
D.
, and
Sanders
,
S. P.
, 2002,
The Physics of Skiing: Skiing at the Triple Point
, 2nd ed.,
Springer
,
New York
.
5.
Reinisch
,
G.
, 1991, “
A Physical Theory of Alpine Ski Racing
,”
Spectrum der Sportwissenschaften
,
3
(
1
), pp.
26
50
.
6.
Shiller
,
Z.
, 1994, “
On Singular Time-Optimal Control Along Specific Paths
,”
IEEE Trans. Rob. Autom.
1042-296X,
10
, pp.
561
566
.
7.
Bryson
,
A. E.
, Jr.
, and
Ho
,
Y. -C.
, 1975,
Applied Optimal Control: Optimization, Estimation, and Control
,
Hemisphere
,
New York
.
8.
Kelley
,
H. J.
,
Kopp
,
R. E.
, and
Moyer
,
H. G.
, 1967, “
Singular Extremals, Topics in Optimization
,”
Mathematics in Science and Engineering
,
G.
Leitmann
, ed.,
,
New York
.
9.
Covic
,
V.
, and
Veskovic
,
M.
, 2008, “
Brachistochrone on a Surface With Coulomb Friction
,”
Int. J. Non-Linear Mech.
0020-7462,
43
, pp.
437
450
.
10.
Wolfram
,
S.
, 2003,
The Mathematica® Book
, 5th ed.,
Wolfram Research
,
Champaign, IL
.
11.
Hanselman
,
D.
, and
Littlefield
,
B.
, 2005,
Mastering MATLAB7
,
Prentice-Hall
,