PROPT The Brachistochrone Problem

This page is part of the PROPT Manual. See PROPT Manual.

This problem was formulated by Johann Bernoulli, in Acta Eruditorum, June 1696

Problem description

"Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time."

In this example, we solve the problem numerically for A = (0,0) and B = (10,-3), and an initial speed of zero.

The mechanical system is modelled as follows:

${\frac {dx}{dt}}=v\sin(\theta )$ ${\frac {dy}{dt}}=-v\cos(\theta )$ ${\frac {dv}{dt}}=g\cos(\theta )$

where (x,y) is the coordinates of the point, v is the velocity, and theta is the angle between the direction of movement and the vertical.

% Copyright (c) 2007-2009 by Tomlab Optimization Inc.

Problem setup

toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 20);
setPhase(p);

tomStates x y v
tomControls theta

% Initial guess
% Note: The guess for t_f must appear in the list before expression involving t.
x0 = {t_f == 10
    icollocate({
    v == t
    x == v*t/2
    y == -1
    })
    collocate(theta==0.1)};

% Box constraints
cbox = {0.1 <= t_f <= 100
    0 <= icollocate(v)
    0 <= collocate(theta) <= pi};

% Boundary constraints
cbnd = {initial({x == 0; y == 0; v == 0})
    final({x == 10; y == -3})};

% ODEs and path constraints
g = 9.81;
ceq = collocate({
    dot(x) == v.*sin(theta)
    dot(y) == -v.*cos(theta)
    dot(v) == g*cos(theta)});

% Objective
objective = t_f;

Solve the problem

options = struct;
options.name = 'Brachistochrone';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
x     = subs(collocate(x),solution);
y     = subs(collocate(y),solution);
v     = subs(collocate(v),solution);
theta = subs(collocate(theta),solution);
t =     subs(collocate(t),solution);

Problem type appears to be: lpcon
Time for symbolic processing: 0.1436 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Brachistochrone                f_k       1.878940329113842700
                                       sum(|constr|)      0.000000174716299412
                              f(x_k) + sum(|constr|)      1.878940503830142100
                                              f(x_0)     10.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv  834 ConJacEv  834 Iter  224 MinorIter  826
CPU time: 0.468003 sec. Elapsed time: 0.467000 sec. 

Plot the result

To obtain the brachistochrone curve, we plot y versus x.

subplot(3,1,1)
plot(x, y);

subplot(3,1,2)
plot(t, v);

% We can also plot theta vs. t.
subplot(3,1,3)
plot(t, theta)