Contrasting Two Transformation-Based Methods for Obtaining Absolute
  Extrema
Contrasting Two Transformation-Based Methods
for Obtaining Absolute Extrema
D. F. M. Torres
and G. Leitmann
Abstract. In this note we contrast two transformation-based methods
to deduce absolute extrema and the corresponding extremizers. Unlike
variation-based methods, the transformation-based ones of Carlson and Leit-
mann and the recent one of Silva and Torres are direct in that they permit
obtaining solutions by inspection.
Mathematics Subject Classification 2000: 49J15, 49M30.
Key Words. Optimal Control, direct methods, Calculus of Variations,
absolute extrema, invariance, symmetry.
1 Introduction
In the mid 1960’s a direct method for the problems of the calculus of vari-
ations, which permits one to obtain absolute extremizers directly, without
using variational methods, was introduced by Leitmann (Ref. 9). Since then,
this direct method has been extended and applied to a variety of problems
(see e.g. Refs. 1, 3, 4, 10). A different but related direct approach to prob-
lems of optimal control, based on the variational symmetries of the problem
(cf. Refs. 7, 8), was recently introduced by Silva and Torres (Ref. 11). The
emphasis in Ref. 11 has been on showing the differences and similarities be-
tween the proposed method and that suggested by Leitmann. In order to
illustrate the relation between these two methods, only examples capable of
1Associate Professor in the Department of Mathematics, University of Aveiro, Aveiro,
Portugal.
2Professor in the Graduate School, College of Engineering, University of California,
Berkeley, California.
http://arxiv.org/abs/0704.0473v1
treatment by both methods were presented in Ref. 11. In this note, we dis-
cuss some differences between the method of Carlson and Leitmann (C/L)
and Silva and Torres (S/T). In particular, we show how one succeeds when
the other does not.
2 The Invariant Transformation Method of S/T3
Let us consider the problem of optimal control in Lagrange form: minimize
an integral
I [x(·), u(·)] =
L (t, x(t), u(t)) dt (1)
subject to a control system
ẋ(t) = ϕ (t, x(t), u(t)) a.e. on [a, b] , (2)
together with appropriate boundary conditions and constraint on the values
of the control variables:
x(a) = xa , x(b) = xb , u(t) ∈ Ω . (3)
The Lagrangian L(·, ·, ·) is a real function assumed to be continuously dif-
ferentiable in [a, b] × Rn × Rm; t ∈ R is the independent variable; x(·) :
[a, b] → Rn the vector of state variables; u(·) : [a, b] → Ω ⊆ Rm the
vector of controls, assumed to be a piecewise continuous function; and
ϕ(·, ·, ·) : [a, b] × Rn × Rm → Rn the derivative function, assumed to be
a continuously differentiable vector function. When Ω is an open set (it
may be all Euclidean space Ω = Rm), problem (1)-(3) can be studied using
the classical techniques of the Calculus of Variations. Optimal Control The-
ory includes the classical Calculus of Variations and generalizes the theory
by dealing with the cases where Ω is not an open set.
The application of the invariant transformation method (Ref. 11) de-
pends on the existence of a sufficiently rich family of invariance transfor-
mations (variational symmetries). The reader interested on the study of
variational symmetries is referred to Refs. 8, 12, 13 and references therein.
Definition 2.1 Let hs be a s-parameter family of C1 mappings satisfying:
hs(·, ·, ·) : [a, b] ×Rn × Ω −→ R× Rn × Rm ,
hs(t, x, u) = (ts(t, x, u), xs(t, x, u), us(t, x, u)) ,
h0(t, x, u) = (t, x, u) , ∀(t, x, u) ∈ [a, b] ×Rn × Ω .
3Throughout this Note, the notation conforms to that used in the references.
If there exists a function Φs(t, x, u) ∈ C1 ([a, b],Rn,Ω;R) such that
L ◦ hs(t, x(t), u(t))
ts (t, x(t), u(t)) = L (t, x(t), u(t)) +
Φs (t, x(t), u(t))
xs (t, x(t), u(t)) = ϕ ◦ hs (t, x(t), u(t))
ts (t, x(t), u(t)) (5)
for all admissible pairs (x(·), u(·)), then (1)–(2) is said to be invariant un-
der the transformations hs(t, x, u) up to Φs(t, x, u); and the transformations
hs(t, x, u) are said to be a variational symmetry of (1)–(2).
The method proposed in Ref. 11 is based on a very simple idea. Given
an optimal control problem, one begins by determining its invariance trans-
formations according to Definition 2.1. With respect to this, the tools devel-
oped in Refs. 7, 8 are useful. Applying the parameter-invariance transfor-
mations, we embed our problem into a parameter-family of optimal control
problems. Given the invariance properties, if we are able to solve one of the
problems of this family, we also get the solution to our original problem (or
to any other problem of the same family) from the invariant transformations.
In section 4 we give an example which shows that the Invariant Transforma-
tion Method (Ref. 11) is more general than the earlier C/L transformation
method in the case of optimal control problems.
3 The Direct Solution Method of C/L
Since this method is fully discussed in readily available references, e.g.
Refs. 1, 3, 4, 9, 10, many in this journal, we shall only recall that the C/L
transformation based method is applicable to problems in the Calculus of
Variations format: minimize an integral
I [x(·)] =
F (t, x(t), ẋ(t)) dt (6)
with given end conditions
x(a) = xa , x(b) = xb . (7)
If one wishes to solve an optimal control problem (1)-(3), the “elimination”
of u(t) in favor of a function of t, x(t), ẋ(t) must be possible. As illustrated
in section 4, this may fail even if the Implicit Function Theorem is satisfied.
Both the S/T and the C/L methods are predicated on posing a problem
“equivalent” to the original problems (1)-(3) and (6)-(7), respectively. Thus,
these methods are useful only if the solution of the “equivalent” problem is
directly obtainable, i.e., by inspection. There is, at present, no result assur-
ing that this can be done in general for the S/T method. However, for the
C/L method, at least in the scalar x case, it has been shown in Ref. 5 and
generalized to open-loop differential games in Ref. 2, that the “equivalent”
problem always has a minimizing solution obtained by inspection. The con-
ditions sufficient for this result are convexity of integrand F (t, x(t), ẋ(t))
with respect to ẋ(t), and existence of a so-called “field of extremals”. In-
deed, no matter what the integrand of the original problem is, provided the
conditions above are met, the absolute minimizer of the equivalent problem
is always a constant.
4 Example 1
The advantage of the invariant transformation method when compared with
the earlier transformation method is that one can apply it directly to control
systems whereas the method of C/L requires that the control u(t) can be
expressed as a sufficiently smooth function of t, x(t), ẋ(t), e.g. such that
the integrand be continuous in x(t) and ẋ(t). Here we use the invariant
transformation method of S/T to solve a simple optimal control problem
that is not covered by the classical theory of the Calculus of Variations and
which can not be solved by the previous transformation method.
Consider the global minimum problem
I[u1(·), u2(·)] =
u1(t)
2 + u2(t)
dt −→ min
ẋ1(t) = exp(u1(t)) + u1(t) + u2(t) ,
ẋ2(t) = u2(t) ,
x1(0) = 0 , x1(1) = 2 , x2(0) = 0 , x2(1) = 1 ,
u1(t) , u2(t) ∈ Ω = [−1, 1] .
We apply the procedure introduced in Ref. 11 and briefly described in sec-
tion 2. First we notice that problem (8) is variationally invariant according
to Definition 2.1 under the one-parameter transformations4
= x1 + st , x
= x2 + st , u
= u2 + s (t
s = t and us
= u1) . (9)
To prove this, we need to show that both the functional integral I[·] and the
control system stay invariant under the s-parameter transformations (9).
This is easily seen by direct calculations. We begin by showing (4):
Is[us
(·), us
(·)] =
+ (us
u1(t)
2 + (u2(t) + s)
u1(t)
2 + u2(t)
s2t+ 2sx2(t)
= I[u1(·), u2(·)] + s
2 + 2s .
We remark that Φs (t, x2) = s
2t+2sx2 and that I
s[·] and I[·] have the same
minimizers: adding a constant s2+2s to the functional I[·] does not change
the minimizer of I[·]. It remains to prove (5):
(t)) =
(x1(t) + st) = ẋ1(t) + s = exp(u1(t)) + u1(t) + u2(t) + s
= exp(us
(t)) + us
(t) + us
(t) ,
(t)) =
(x2(t) + st) = ẋ2(t) + s = u2(t) + s
(t) .
Conditions (10) and (11) prove that problem (8) is invariant under the
s-parameter transformations (9) up to the gauge term Φs = s2t + 2sx2.
Using the invariance transformations (9), we generalize problem (8) to a
s-parameter family of problems, s ∈ R, which include the original problem
4A computer algebra package that can be used to find the invariance trans-
formations (see Refs. 7, 8) is available from the Maple Application Center at
http://www.maplesoft.com/applications/app center view.aspx?AID=1983
for s = 0:
Is[us
(·), us
(·)] =
(t))2 + (us
(t))2dt −→ min
(t) = exp(us
(t)) + us
(t) + us
(t) ,
ẋ2(t) = u
(t) ,
(0) = 0 , xs
(1) = 2 + s , xs
(0) = 0 , xs
(1) = 1 + s ,
(t) ∈ [−1, 1] , us
(t) ∈ [−1 + s, 1 + s] .
It is clear that Is ≥ 0 and that Is = 0 if us
(t) = us
(t) ≡ 0. The con-
trol equation, the boundary conditions and the constraints on the values of
the controls, imply that us
(t) = us
(t) ≡ 0 is admissible only if s = −1:
xs=−1
(t) = t, xs=−1
(t) ≡ 0. Hence, for s = −1 the global minimum to Is[·]
is 0 and the minimizing trajectory is given by
(t) ≡ 0 , ũs
(t) ≡ 0 , x̃s
(t) = t , x̃s
(t) ≡ 0 .
Since for any s one has by (10) that I[u1(·), u2(·)] = I
(·), us
(·)]−s2−2s,
we conclude that the global minimum for problem I[u1(·), u2(·)] is 1. Thus,
using the inverse functions of the variational symmetries (9),
u1(t) = u
(t) , u2(t) = u
(t)−s , x1(t) = x
(t)−st , x2(t) = x
(t)−st ,
the absolute minimizer is
ũ1(t) = 0 , ũ2(t) = 1 , x̃1(t) = 2t , x̃2(t) = t .
This problem cannot be solved by the C/L method.
While the existence of a function h(·) such that u1(t) = h (ẋ1(t)− ẋ2(t))
is assured by the satisfaction of the Implicit Function Theorem, this is not
useful in “eliminating” the control in favor of t, x(t), ẋ(t) which is a re-
quirement of the C/L method. This is so because h(·) is a solution of a
transcendental equation. In addition, since the controls are bounded, even
if “elimination” of the control were possible, this would lead to differential
constraints of the form briefly discussed in Ref. 6.
5 Example 2
Consider the problem of attaining the absolute minimum of integral
I[x(·)] =
ẋ2(t) + tẋ(t)
dt (12)
with prescribed end conditions
x(a) = xa , x(b) = xb . (13)
This is a problem in the Calculus of Variations format for which the C/L or
S/T transformation-based method applies. These methods can be applied ab
initio to obtain the solution. However, here we shall employ the constructive
sufficiency condition embodied in Theorem 7 of Ref. 5 towards that end.
The Euler-Lagrange equation is
ẍ(t) = −
so that
x(t) = −
t2 + c1t+ c2 (15)
is the extremal with the constants of integration given by end conditions
(13), say ci = c
, i = 1, 2, and the extremal satisfying (15) is
x∗(t) = −
t2 + c∗
t+ c∗
which, being the solution of necessary condition (14), is a candidate for the
absolute minimizer of (12)-(13).
Now we can readily show that the conditions of Theorem 7 of Ref. 5 are
met. Consider the one-parameter family of extremals
ξ(t, β) = −
t2 + c∗
t+ c∗
+ β .
First of all, there exist a β∗, namely β∗ = 0, such that
ξ(t, β∗) = x∗(t) .
Secondly,
∂ξ(t, β)
6= 0 ,
and finally, the integrand of (12) is convex in ẋ(t).
Thus, the extremal (16) is indeed the absolute minimizer of (12)-(13).
Of course, this is precisely the solution obtained by employing the C/L
method. Indeed, the more general method inherent in Theorem 7 of Ref. 5
was derived using the C/L transformation-based method.
6 Conclusion
We have contrasted two transformation-based methods for obtaining abso-
lutely extremizing solutions for two classes of problems. One, dubbed the
Carlson/Leitmann method, is applicable to problems in the format of the
Calculus of Variations. The other, due to Silva and Torres, is applicable to
problems of Optimal Control.
We have shown that it is not always possible to convert an Optimal
Control problem into a Calculus of Variations one. Hence, the C/L method
may fail to apply when the S/T succeeds. On the other hand, a classical
constructive sufficiency condition, readily derived by the C/L method, ren-
ders absolute extremizers for specific problems of the Calculus of Variations
more directly and succinctly than the C/L and S/T methods.
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	Introduction
	The Invariant Transformation Method of S/T3
	The Direct Solution Method of C/L
	Example 1
	Example 2
	Conclusion
