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Classical Problems in Calculus of Variations
and Optimal Control
Stephen Lamb
Supervised by Lyle Noakes
The University of Western Australia
Vacation Research Scholarships are funded jointly by the Department of Education and Training
and the Australian Mathematical Sciences Institute.
1 Introduction
Calculus of variations (COV) is a field of mathematics that deals with finding the extremals of
Lagrangian functions defined by functionals, in an attempt to find optimal solutions. Although the
subject has a long and rich history, current research in the field is still producing new results. Op-
timal control is the generalisation of the calculus of variations. My project focuses on the classical
problem of COV, and how the tools of optimal control can be used to simplify results, and even
produce results where COV was too blunt an object. My project started with a brief study of some
classical problems in COV and then I started generating the differential equations that we solve to
answer the problems. These include such problems as the brachistochrone and the catenary.
The next task was to learn about optimal control and its link to COV. Once able to write out
our control problem, I needed a similar tool to solve for extremals as above, which in optimal con-
trol is the Pontryagin Maximum Principle. The most crucial part of the project was about learning
and understanding how to use the Pontryagin Maximum Principle to solve the optimal control
problems. We then extended it from Rn to other smooth manifolds using Riemannian geometry.
Weusedourframeworkofoptimalcontrol to solve two very important classical problems: geodesics
on Riemannian manifolds and elastica curves. Lastly, after generating solutions to these problems
for different spaces, we had to have a method of solving these boundary value problems numerically,
for problems where no closed form solution exists.
The study of classical problems in calculus of variations and optimal control has provided me with
a good foundation for further study into differential geometry, optimisation methods and functional
analysis. It has also given good insight into the topics of topology, Riemannian geometry and anal-
ysis. Although I have aquired a deeper level of understanding through this research project, I am
left with far more questions to answer.
Future work into the topic involves study of dependence of these classical problems to assumed
conditions. e.g. The study of the brachistochrone under different gravitational field conditions or
the catenary under a changing viscocity of air as a function of height. Further research into geodesics
phrased using optimal control could involve looking into the geodesics on manifolds that aren’t as
nice and symmetric as the ones chosen. Also research into the existence of abnormal geodesics
in sub-Riemannian manifolds would yield enlightening results. For the problems of elastic curves,
future work could involve the study of elastic curves in other riemannian manifolds. Lastly further
investigation into other numerical solving methods for solving these problems, aiming at reducing
computing cost could prove a good idea.
2
2 Calculus of Variations
To understand what calculus of variations is, and in turn what optimal control is, we require
understanding of the lagrangian function and how to determine extremals from it.
Definition 2.1. The Lagrangian (L) is an energy function defined on the Tangent bundle, that
maps onto the space of real numbers. We can write L as L(q(t),q˙(t))
L:TM→R
Calculus of variations is concerned with finding the extremals of functions defined by:
Z t
2
J(t) = L(q(t),q˙(t))dt, (1)
t1
where extremals are the critical points of the Lagrangian.
Theorem 2.2. If q is an extremal of the Lagrangian, then it must satisfy the Euler-Lagrange
equation:
d ∂L = ∂L (2)
dt ∂q˙ ∂q
Proof. Consider the system defined above by (1), and let’s assume that q is an extremal. Without
loss of generality, we will assume that q is a minimising function. Hence we can conclude that:
L(q(t),q˙(t)) < L(q(t) + ǫη(t),q˙(t)ǫη˙(t))
Therefore if we take the derivative with respect to ǫ, the following result is obtained:
Z t
d d 2
J(t)| = L(q(t)+ǫη(t),q˙(t)+ǫη˙(t))dt| =0
dǫ ǫ=0 dǫ ǫ=0
t
1
=⇒ Z t2∂Lη+ ∂Lη˙dt=0
t1 ∂q ∂q˙
Using integration by parts on the second term above, we get the following:
Z t
2 ∂L d ∂L
=⇒ t ∂qη−dt ∂q˙η dt=0
1
Factoring out η we get the Euler-Lagrange equation:
d ∂L= ∂L
dt ∂q˙ ∂q
3
2.1 Classical Problems in COV
2.1.1 Brachistochrone
The Brachistochrone is a classical problem that deals with finding a curve between two points:
(x ,y ) and (x ,y ) with x < x and y < y , such that the time taken for a bead to go along the
0 0 1 1 0 1 0 1
curve between the two points on the curve is minimal.
Time is the variable that is going to be minimised.
Let the total length of the curve be L. Using Newtons equations of motion, we can derive the
equation for total time taken to traverse the curve:
T(y) = Z 0 ds ds (3)
L v(s)
(Where v(s) is the velocity and ds is the arclength)
Using the conservation of energy principle, we can define this expression in terms of y(x).
KE+PE=E=constant
E=mgy(x)+1mv(x)2 (4)
2
Rearranging for v(x), we get the following:
v(x) = r2(E −mgy(x)) (5)
m
Thus using (3), we get the following result:
Z x p
1 1+y˙2
T(y) = x0 q2(E−mgy(x))dx (6)
m
With y(x ) = y , and y(x ) = y
0 0 1 1
Simplifying this expression with the substitution: z = 1 2E−2gmy(x) we get the following func-
2g m
tional: Z r
p x1 1+z˙2
J(z) = 2g z dx (7)
x0
Using the Euler-Lagrange equation, we derive the following differential equation:
z(1+z˙2) = c (8)
Using the substitution z = tan(θ), then 1+z˙2 = sec(θ). Hence the curve that satisfies the conditions
of the brachistochrone curve is the parametric equation of a cycloid:
y(θ) = d1(1+cos(2θ)) (9)
x(θ) = d −d (2θ+sin(2θ)) (10)
2 1
4
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