Necessary optimality conditions for the calculus of variations on time
  scales
Necessary optimality conditions for the calculus
of variations on time scales∗
Rui A. C. Ferreira
ruiacferreira@yahoo.com
Delfim F. M. Torres
delfim@mat.ua.pt
Department of Mathematics
University of Aveiro
3810-193 Aveiro, Portugal
Abstract
We study more general variational problems on time scales. Previous
results are generalized by proving necessary optimality conditions for (i)
variational problems involving delta derivatives of more than the first
order, and (ii) problems of the calculus of variations with delta-differential
side conditions (Lagrange problem of the calculus of variations on time
scales).
Keywords: time scales, ∆-variational calculus, higher-order ∆-derivatives,
higher-order Euler-Lagrange ∆-equations, Lagrange problem on time scales,
normal and abnormal ∆-extremals.
2000 Mathematics Subject Classification: 49K05, 39A12.
1 Introduction
The theory of time scales is a relatively new area, that unify and generalize
difference and differential equations [5]. It was initiated by Stefan Hilger in the
nineties of the XX century [7, 8], and is now subject of strong current research in
many different fields in which dynamic processes can be described with discrete
or continuous models [1].
The calculus of variations on time scales was introduced by Bohner [4] and by
Hilscher and Zeidan [9], and appears to have many opportunities for application
in economics [2]. In all those works, necessary optimality conditions are only
obtained for the basic (simplest) problem of the calculus of variations on time
scales: in [2, 4] for the basic problem with fixed endpoints, in [9] for the basic
problem with general (jointly varying) endpoints. Having in mind the classical
∗This work is part of the first author’s PhD project.
http://arxiv.org/abs/0704.0656v1
setting (situation when the time scale T is either R or Z – see e.g. [6, 14] and
[10, 11], respectively), one suspects that the Euler-Lagrange equations in [2, 4, 9]
are easily generalized for problems with higher-order delta derivatives. This is
not exactly the case, even beginning with the formulation of the problem.
The basic problem of the calculus of variations on time scales is defined (cf.
[4, 9], see §2 below for the meaning of the ∆-derivative and ∆-integral) as
L[y(·)] =
L(t, yσ(t), y∆(t))∆t −→ min, (y(a) = ya) , (y(b) = yb) , (1)
with L : T×Rn×Rn → R, (y, u) → L(t, y, u) a C2-function for each t, and where
we are using parentheses around the endpoint conditions as a notation to mean
that the conditions may or may not be present: the case with fixed boundary
conditions y(a) = ya and y(b) = yb is studied in [4], for admissible functions y(·)
belonging to C1rd (T;R
n) (rd-continuously ∆-differentiable functions); general
boundary conditions of the type f(y(a), y(b)) = 0, which include the case y(a) or
y(b) free, and over admissible functions in the wider class C1prd (T;R
n) (piecewise
rd-continuously ∆-differentiable functions), are considered in [9]. One question
immediately comes to mind. Why is the basic problem on time scales defined
as (1) and not as
L[y(·)] =
L(t, y(t), y∆(t))∆t −→ min, (y(a) = ya) , (y(b) = yb) . (2)
The answer is simple: compared with (2), definition (1) simplifies the Euler-
Lagrange equation, in the sense that makes it similar to the classical context.
The reader is invited to compare the Euler-Lagrange condition (6) of problem
(1) and the Euler-Lagrange condition (13) of problem (2), with the classical
expression (on the time scale T = R):
Ly′(t, y∗(t), y
(t)) = Ly(t, y∗(t), y
(t)), t ∈ [a, b] .
It turns out that problems (1) and (2) are equivalent: as far as we are assuming
y(·) to be ∆-differentiable, then y(t) = yσ(t) − µ(t)y∆(t) and (i) any problem
(1) can be written in the form (2), (ii) any problem (2) can be written in the
form (1). We claim, however, that the formulation (2) we are promoting here is
more natural and convenient. An advantage of our formulation (2) with respect
to (1) is that it makes clear how to generalize the basic problem on time scales
to the case of a Lagrangian L containing delta derivatives of y(·) up to an order
r, r ≥ 1. The higher-order problem will be naturally defined as
L[y(·)] =
r−1(b)
L(t, y(t), y∆(t), . . . , y∆
(t))∆t −→ min,
y(a) = y0a
r−1(b)
= y0b
, (3)
(a) = yr−1a
r−1 (
ρr−1(b)
= yr−1
where y∆
(t) ∈ Rn, i ∈ {0, . . . , r}, y∆
= y, and n, r ∈ N (assumptions on
the data of the problem will be specified later, in Section 3). One of the new
results in this paper is a necessary optimality condition in delta integral form
for problem (3) (Theorem 4). It is obtained using the interplay of problems (1)
and (2) in order to deal with more general optimal control problems (16).
The paper is organized as follows. In Section 2 we give a brief introduction
to time scales and recall the main results of the calculus of variations on this
general setting. Our contributions are found in Section 3. We start in §3.1
by proving the Euler-Lagrange equation and transversality conditions (natural
boundary conditions – y(a) or/and y(b) free) for the basic problem (2) (The-
orem 2). As a corollary, the Euler-Lagrange equation in [4] and [9] for (1) is
obtained. Regarding the natural boundary conditions, the one which appears
when y(a) is free turns out to be simpler and more close in aspect to the classical
condition Ly′(a, y∗(a), y
(a)) = 0 for problem (1) than to (2)—compare condi-
tion (9) for problem (2) with the correspondent condition (14) for problem (1);
but the inverse situation happens when y(b) is free—compare condition (15) for
problem (1) with the correspondent condition (10) for (2), this last being simpler
and more close in aspect to the classical expression Ly′(b, y∗(b), y
(b)) = 0 valid
on the time scale T = R. In §3.2 we formulate a more general optimal control
problem (16) on time scales, proving respective necessary optimality conditions
in Hamiltonian form (Theorem 3). As corollaries, we obtain a Lagrange multi-
plier rule on time-scales (Corollary 2), and in §3.3 the Euler-Lagrange equation
for the problem of the calculus of variations with higher order delta derivatives
(Theorem 4). Finally, as an illustrative example, we consider in §4 a discrete
time scale and obtain the well-known Euler-Lagrange equation in delta differ-
entiated form.
All the results obtained in this paper can be extended: (i) to nabla deriva-
tives (see [5, §8.4]) with the appropriate modifications and as done in [2] for the
simplest functional; (ii) to more general classes of admissible functions and to
problems with more general boundary conditions, as done in [9] for the simplest
functional of the calculus of variations on time scales.
2 Time scales and previous results
We begin by recalling the main definitions and properties of time scales (cf.
[1, 5, 7, 8] and references therein).
A nonempty closed subset of R is called a Time Scale and is denoted by T.
The forward jump operator σ : T → T is defined by
σ(t) = inf {s ∈ T : s > t}, for all t ∈ T,
while the backward jump operator ρ : T → T is defined by
ρ(t) = sup {s ∈ T : s < t}, for all t ∈ T,
with inf ∅ = supT (i.e., σ(M) = M if T has a maximum M) and sup ∅ = inf T
(i.e., ρ(m) = m if T has a minimum m).
A point t ∈ T is called right-dense, right-scattered, left-dense and left-
scattered if σ(t) = t, σ(t) > t, ρ(t) = t and ρ(t) < t, respectively.
Throughout the text we let T = [a, b] ∩ T0 with a < b and T0 a time scale.
We define Tk = T\(ρ(b), b], Tk
and more generally Tk
for n ∈ N. The following standard notation is used for σ (and ρ): σ0(t) = t,
σn(t) = (σ ◦ σn−1)(t), n ∈ N.
The graininess function µ : T → [0,∞) is defined by
µ(t) = σ(t)− t, for all t ∈ T.
We say that a function f : T → R is delta differentiable at t ∈ Tk if there is
a number f∆(t) such that for all ε > 0 there exists a neighborhood U of t (i.e.,
U = (t− δ, t+ δ) ∩ T for some δ > 0) such that
|f(σ(t)) − f(s)− f∆(t)(σ(t) − s)| ≤ ε|σ(t)− s|, for all s ∈ U.
We call f∆(t) the delta derivative of f at t.
Now, we define the rth−delta derivative (r ∈ N) of f to be the function
→ R, provided f∆
is delta differentiable on Tk
For delta differentiable f and g, the next formulas hold:
fσ(t) = f(t) + µ(t)f∆(t) (4)
(fg)∆(t) = f∆(t)gσ(t) + f(t)g∆(t)
= f∆(t)g(t) + fσ(t)g∆(t),
where we abbreviate f ◦ σ by fσ.
Next, a function f : T → R is called rd-continuous if it is continuous at
right-dense points and if its left-sided limit exists at left-dense points. We
denote the set of all rd-continuous functions by Crd or Crd[T] and the set of all
delta differentiable functions with rd-continuous derivative by C1rd or C
rd[T].
It is known that rd-continuous functions possess an antiderivative, i.e., there
exists a function F with F∆ = f , and in this case an integral is defined by
f(t)∆t = F (b)− F (a). It satisfies
∫ σ(t)
f(τ)∆τ = µ(t)f(t) . (5)
We now present some useful properties of the delta integral:
Lemma 1. If a, b ∈ T and f, g ∈Crd, then
f(σ(t))g∆(t)∆t = [(fg)(t)]
t=a −
f∆(t)g(t)∆t.
f(t)g∆(t)∆t = [(fg)(t)]
t=a −
f∆(t)g(σ(t))∆t.
The main result of the calculus of variations on time scales is given by the
following necessary optimality condition for problem (1).
Theorem 1 ([4]). If y∗ is a weak local minimizer (cf. §3) of the problem
L[y(·)] =
L(t, yσ(t), y∆(t))∆t −→ min
y(·) ∈ C1
y(a) = ya, y(b) = yb,
then the Euler-Lagrange equation
L∆y∆(t, y
(t), y∆
(t)) = Lyσ(t, y
(t), y∆
(t)), t ∈ Tk
holds.
Main ingredients to prove Theorem 1 are item 1 of Lemma 1 and the Dubois-
Reymond lemma:
Lemma 2 ([4]). Let g ∈ Crd, g : [a, b]
k → Rn. Then,
g(t) · η∆(t)∆t = 0 for all η ∈ C1
with η(a) = η(b) = 0
if and only if
g(t) = c on [a, b]k for some c ∈ Rn.
3 Main results
Assume that the Lagrangian L(t, u0(t), u1(t), . . . , ur(t)) (r ≥ 1) is a C
r+1 func-
tion of (u0(t), u1(t), . . . , ur(t)) for each t ∈ T. Let y ∈ C
rd[T], where
Crrd[T] =
y : Tk
→ Rn : y∆
is rd-continuous on Tk
We want to minimize the functional L of problem (3). For this, we say that
y∗ ∈ C
rd[T] is a weak local minimizer for the variational problem (3) provided
there exists δ > 0 such that L[y∗] ≤ L[y] for all y ∈ C
rd[T] satisfying the
constraints in (3) and ‖y − y∗‖r,∞ < δ, where
||y||r,∞ :=
with y∆
= y and ||y||∞ := supt∈Tkr |y(t)|.
3.1 The basic problem on time scales
We start by proving the necessary optimality condition for the simplest varia-
tional problem (r = 1):
L[y(·)] =
L(t, y(t), y∆(t))∆t −→ min
y(·) ∈ C1rd[T]
(y(a) = ya) , (y(b) = yb) .
Remark 1. We are assuming in problem (7) that the time scale T has at least
3 points. Indeed, for the delta-integral to be defined we need at least 2 points.
Assume that the time scale has only two points: T = {a, b}, with b = σ(a).
Then,
∫ σ(a)
L(t, y(t), y∆(t))∆t = µ(a)L(a, y(a), y∆(a)). In the case both y(a)
and y(σ(a)) are fixed, since y∆(a) =
y(σ(a))−y(a)
, then L[y(·)] would be a con-
stant for every admissible function y(·) (there would be nothing to minimize
and problem (7) would be trivial). Similarly, for (3) we assume the time scale
to have at least 2r + 1 points (see Remark 15).
Theorem 2. If y∗ is a weak local minimizer of (7) (problem (3) with r = 1),
then the Euler-Lagrange equation in ∆-integral form
Ly∆(t, y∗(t), y
(t)) =
∫ σ(t)
Ly(ξ, y∗(ξ), y
(ξ))∆ξ + c (8)
holds ∀t ∈ Tk and some c ∈ Rn. Moreover, if the initial condition y(a) = ya is
not present (y(a) is free), then the supplementary condition
Ly∆(a, y∗(a), y
(a))− µ(a)Ly(a, y∗(a), y
(a)) = 0 (9)
holds; if y(b) = yb is not present (y(b) is free), then
Ly∆(ρ(b), y∗(ρ(b)), y
(ρ(b))) = 0 . (10)
Remark 2. For the time scale T = R equalities (9) and (10) give, respec-
tively, the well-known natural boundary conditions Ly′(a, y∗(a), y
(a)) = 0 and
Ly′(b, y∗(b), y
(b)) = 0.
Proof. Suppose that y∗(·) is a weak local minimizer of L[·]. Let η(·) ∈C
rd and
define Φ : R → R by
Φ(ε) = L[y∗(·) + εη(·)].
This function has a minimum at ε = 0, so we must have Φ′(0) = 0. Applying
the delta-integral properties and the integration by parts formula 2 (second item
in Lemma 1), we have
0 = Φ′(0)
[Ly(t, y∗(t), y
(t)) · η(t) + Ly∆(t, y∗(t), y
(t)) · η∆(t)]∆t
Ly(t, y∗(t), y
(t)∆t · η(t)
∫ σ(t)
Ly(ξ, y∗(ξ), y
(ξ))∆ξ · η∆(t)− Ly∆(t, y∗(t), y
(t)) · η∆(t)
Let us limit the set of all delta-differentiable functions η(·) with rd-continuous
derivatives to those which satisfy the condition η(a) = η(b) = 0 (this condition
is satisfied by all the admissible variations η(·) in the case both y(a) = ya and
y(b) = yb are fixed). For these functions we have
Ly∆(t, y∗(t), y
(t)) −
∫ σ(t)
Ly(ξ, y∗(ξ), y
(ξ))∆ξ
· η∆(t)∆t = 0 .
Therefore, by the lemma of Dubois-Reymond (Lemma 2), there exists a constant
c ∈ Rn such that (8) holds:
Ly∆(t, y∗(t), y
(t))−
∫ σ(t)
Ly(ξ, y∗(ξ), y
(ξ))∆ξ = c , (12)
for all t ∈ Tk. Because of (12), condition (11) simplifies to
Ly(t, y∗(t), y
(t)∆t · η(t)
+ c · η(t)|
t=a = 0,
for any admissible η(·). If y(a) = ya is not present in problem (7) (so that η(a)
need not to be zero), taking η(t) = t− b we find that c = 0; if y(b) = yb is not
present, taking η(t) = t − a we find that
Ly(t, y∗(t), y
(t) = 0. Applying
these two conditions to (12) and having in mind formula (5), we may state that
Ly∆(a, y∗(a), y
(a))−
∫ σ(a)
Ly(ξ, y∗(ξ), y
(ξ))∆ξ = 0
⇔ Ly∆(a, y∗(a), y
(a))− µ(a)Ly(a, y∗(a), y
(a)) = 0,
and (note that σ(ρ(b)) = b)
Ly∆(ρ(b), y∗(ρ(b)), y
(ρ(b)))−
Ly(ξ, y∗(ξ), y
(ξ))∆ξ = 0
⇔ Ly∆(ρ(b), y∗(ρ(b)), y
(ρ(b))) = 0.
Remark 3. Since σ(t) ≥ t, ∀t ∈ T, we must have
Ly∆(t, y∗(t), y
(t))−
∫ σ(t)
Ly(ξ, y∗(ξ), y
(ξ))∆ξ = c
⇔ Ly∆(t, y∗(t), y
(t)) − µ(t)Ly(t, y∗(t), y
Ly(ξ, y∗(ξ), y
(ξ))∆ξ + c,
by formula (5). Delta differentiating both sides, we obtain
Ly∆(t, y∗(t), y
(t))− µ(t)Ly(t, y∗(t), y
= Ly(t, y∗(t), y
(t)), t ∈ Tk
. (13)
Note that we can’t expand the left hand side of this last equation, because we
are not assuming that µ(t) is delta differentiable. In fact, generally µ(t) is not
delta differentiable (see example 1.55, page 21 of [5]). We say that (13) is the
Euler-Lagrange equation for problem (7) in the delta differentiated form.
As mentioned in the introduction, the formulations of the problems of the cal-
culus of variations on time scales with “
t, yσ(t), y∆(t)
” and with “
t, y(t), y∆(t)
are equivalent. It is trivial to derive previous Euler-Lagrange equation (6) from
our equation (13) and the other way around (one can derive (13) directly from
(6)).
Corollary 1. If y∗ ∈ C
[T] is a weak local minimizer of
L[y(·)] =
L(t, yσ(t), y∆(t))∆t , (y(a) = ya), (y(b) = yb),
then the Euler-Lagrange equation (6) holds. If y(a) is free, then the extra
transversality condition (natural boundary condition)
Ly∆(a, y
(a), y∆
(a)) = 0 (14)
holds; if y(b) is free, then
Lyσ(ρ(b), y
(ρ(b)), y∆
(ρ(b)))µ(ρ(b)) + Ly∆(ρ(b), y
(ρ(b)), y∆
(ρ(b))) = 0 . (15)
Proof. Since y(t) is delta differentiable, then (4) holds. This permits us to write
L(t, yσ(t), y∆(t)) = L(t, y(t) + µ(t)y∆(t), y∆(t)) = F (t, y(t), y∆(t)).
Applying equation (13) to the functional F we obtain
Fy∆(t, y(t), y
∆(t)) − µ(t)Fy(t, y(t), y
∆(t))
= Fy(t, y(t), y
∆(t)).
Fy(t, y(t), y
∆(t)) = Lyσ(t, y
σ(t), y∆(t)) ,
Fy∆(t, y(t), y
∆(t)) = Lyσ(t, y
σ(t), y∆(t))µ(t) + Ly∆(t, y
σ(t), y∆(t)) ,
and the result follows.
3.2 The Lagrange problem on time scales
Now we consider a more general variational problem with delta-differential side
conditions:
J [y(·), u(·)] =
L(t, y(t), u(t))∆t −→ min ,
y∆(t) = ϕ(t, y(t), u(t)) ,
(y(a) = ya) , (y(b) = yb) ,
where y(·) ∈ C1rd[T], u(·) ∈ Crd[T], y(t) ∈ R
n and u(t) ∈ Rm for all t ∈ T,
and m ≤ n. We assume L : T × Rn × Rm → R and ϕ : T × Rn × Rm → Rn
to be C1-functions of y and u for each t; and that for each control function
u(·) ∈ Crd[T;R
m] there exists a correspondent y(·) ∈ C1rd[T;R
n] solution of
the ∆-differential equation y∆(t) = ϕ(t, y(t), u(t)). We remark that conditions
for existence or uniqueness are available for O∆E’s from the very beginning
of the theory of time scales (see [8, Theorem 8]). Roughly speaking, forward
solutions exist, while existence of backward solutions needs extra assumptions
(e.g. regressivity). In control theory, however, one usually needs only forward
solutions, so we do not need to impose such extra assumptions [3].
We are interested to find necessary conditions for a pair (y∗, u∗) to be a weak
local minimizer of J .
Definition 1. Take an admissible pair (y∗, u∗). We say that (y∗, u∗) is a weak
local minimizer for (16) if there exist δ > 0 such that J [y∗, u∗] ≤ J [y, u] for all
admissible pairs (y, u) satisfying ‖y − y∗‖1,∞ + ‖u− u∗‖∞ < δ.
Remark 4. Problem (16) is very general and includes: (i) problem (7) (this
is the particular case where m = n and ϕ(t, y, u) = u), (ii) the problem of
the calculus of variations with higher-order delta derivatives (3) (such problem
receive special attention in Section 3.3 below), (iii) isoperimetric problems on
time scales. Suppose that the isoperimetric condition
I[y(·), u(·)] =
g (t, y(t), u(t))∆t = β ,
β a given constant, is prescribed. We can introduce a new state variable yn+1
defined by
yn+1(t) =
g(ξ, y(ξ), u(ξ))∆ξ, t ∈ T,
with boundary conditions yn+1(a) = 0 and yn+1(b) = β. Then
y∆n+1(t) = g (t, y(t), u(t)) , t ∈ T
and we can always recast an isoperimetric problem as a Lagrange problem (16).
To establish necessary optimality conditions for (16) is more complicated
than for the basic problem of the calculus of variations on time scales (1) or (2),
owing to the possibility of existence of abnormal extremals (Definition 2). The
abnormal case never occurs for the basic problem (Proposition 2).
Theorem 3 (The weak maximum principle on time scales). If (y∗(·), u∗(·)) is
a weak local minimizer of problem (16), then there exists a set of multipliers
(ψ0∗ , ψ∗(·)) 6= 0, where ψ0∗ is a nonnegative constant and ψ∗(·) : T → R
n is a
delta differentiable function on Tk, such that (y∗(·), u∗(·), ψ0∗ , ψ∗(·)) satisfy
(t) = Hψσ (t, y∗(t), u∗(t), ψ0∗ , ψ
(t)) , (∆-dynamic equation for y) (17)
(t) = −Hy(t, y∗(t), u∗(t), ψ0∗ , ψ
(t)) , (∆-dynamic equation for ψ) (18)
Hu(t, y∗(t), u∗(t), ψ0∗ , ψ
(t)) = 0 , (∆-stationary condition) (19)
for all t ∈ Tk, where the Hamiltonian function H is defined by
H(t, y, u, ψ0, ψ
σ) = ψ0L(t, y, u) + ψ
σ · ϕ(t, y, u) . (20)
If y(a) is free in (16), then
ψ∗(a) = 0 ; (21)
if y(b) is free in (16), then
ψ∗(b) = 0 . (22)
Remark 5. From the definition (20) of H , it follows immediately that (17) holds
true for any admissible pair (y(·), u(·)) of problem (16). Indeed, condition (17)
is nothing more than the control system y∆
(t) = ϕ(t, y∗(t), u∗(t)).
Remark 6. For the time scale T = Z, (17)-(19) reduce to well-known conditions
in discrete time (see e.g. [13, Ch. 8]): the ∆-dynamic equation for y takes the
form y(k + 1) − y(k) = Hψ (k, y(k), u(k), ψ0, ψ(k + 1)); the ∆-dynamic equa-
tion for ψ gives ψ(k + 1) − ψ(k) = −Hy (k, y(k), u(k), ψ0, ψ(k + 1)); and the
∆-stationary condition reads as Hu (k, y(k), u(k), ψ0, ψ(k + 1)) = 0; with the
Hamiltonian H = ψ0L(k, y(k), u(k)) + ψ(k + 1) · ϕ(k, y(k), u(k)). For T = R,
Theorem 3 is known in the literature as Hestenes necessary condition, which is
a particular case of the Pontryagin Maximum Principle [12].
Corollary 2 (Lagrange multiplier rule on time scales). If (y∗(·), u∗(·)) is a
weak local minimizer of problem (16), then there exists a collection of multipli-
ers (ψ0∗ , ψ∗(·)), ψ0∗ a nonnegative constant and ψ∗(·) : T → R
n a delta dif-
ferentiable function on Tk, not all vanishing, such that (y∗(·), u∗(·), ψ0∗ , ψ∗(·))
satisfy the Euler-Lagrange equation of the augmented functional J∗:
J∗[y(·),u(·), ψ(·)] =
t, y(t), u(t), ψσ(t), y∆(t)
ψ0L(t, y(t), u(t)) + ψ
σ(t) ·
ϕ(t, y(t), u(t)) − y∆(t)
[H(t, y(t), u(t), ψ0, ψ
σ(t))− ψσ(t) · y∆(t)]∆t .
Proof. The Euler-Lagrange equations (13) and (6) applied to (23) give
y∆ − µ(t)L
= L∗y ,
(−µ(t)L∗u)
= L∗u , L
ψσ = 0 ,
that is,
(ψσ(t) + µ(t) ·Hy)
= −Hy , (24)
(−µ(t)Hu)
∆ = Hu , (25)
∆(t) = Hψσ ,
where the partial derivatives ofH are evaluated at (t, y(t), u(t), ψ0, ψ
σ(t)). Obvi-
ously, from (19) we obtain (25). It remains to prove that (18) implies (24) along
(y∗(·), u∗(·), ψ0∗ , ψ∗(·)). Indeed, from (18) we can write µ(t)ψ
∆(t) = −µ(t)Hy,
which is equivalent to ψ(t) = ψσ(t) + µ(t)Hy.
Remark 7. Condition (18) or (24) imply that along the minimizer
ψσ(t) = −
∫ σ(t)
Hy(ξ, y(ξ), u(ξ), ψ0, ψ
σ(ξ))∆ξ − c (26)
for some c ∈ Rn.
Remark 8. The assertion in Theorem 3 that the multipliers cannot be all zero is
crucial. Indeed, without this requirement, for any admissible pair (y(·), u(·)) of
(16) there would always exist a set of multipliers satisfying (18)-(19) (namely,
ψ0 = 0 and ψ(t) ≡ 0).
Remark 9. Along all the work we consider ψ as a row-vector.
Remark 10. If the multipliers (ψ0, ψ(·)) satisfy the conditions of Theorem 3,
then (γψ0, γψ(·)) also do, for any γ > 0. This simple observation allow us to
conclude that it is enough to consider two cases: ψ0 = 0 or ψ0 = 1.
Definition 2. An admissible quadruple (y(·), u(·), ψ0, ψ(·)) satisfying condi-
tions (17)-(19) (also (21) or (22) if y(a) or y(b) are, respectively, free) is called
an extremal for problem (16). An extremal is said to be normal if ψ0 = 1 and
abnormal if ψ0 = 0.
So, Theorem 3 asserts that every minimizer is an extremal.
Proposition 1. The Lagrange problem on time scales (16) has no abnormal
extremals (in particular, all the minimizers are normal) when at least one of the
boundary conditions y(a) or y(b) is absent (when y(a) or y(b) is free).
Proof. Without loss of generality, let us consider y(b) free. We want to prove
that the nonnegative constant ψ0 is nonzero. The fact that ψ0 6= 0 follows from
Theorem 3. Indeed, the multipliers ψ0 and ψ(t) cannot vanish simultaneously
at any point of t ∈ T. As far as y(b) is free, the solution to the problem must
satisfy the condition ψ(b) = 0. The condition ψ(b) = 0 requires a nonzero value
for ψ0 at t = b. But since ψ0 is a nonnegative constant, we conclude that ψ0 is
positive, and we can normalize it (Remark 10) to unity.
Remark 11. In the general situation abnormal extremals may occur. More
precisely (see proof of Theorem 3), abnormality is characterized by the existence
of a nontrivial solution ψ(t) for the system ψ∆(t) + ψσ(t) · ϕy = 0.
Proposition 2. There are no abnormal extremals for problem (7), even in the
case y(a) and y(b) are both fixed (y(a) = ya, y(b) = yb).
Proof. Problem (7) is the particular case of (16) with y∆(t) = u(t). If ψ0 = 0,
then the Hamiltonian (20) takes the form H = ψσ · u. From Theorem 3, ψ∆ = 0
and ψσ = 0, for all t ∈ Tk. Since ψσ = ψ + µ(t)ψ∆, this means that ψ0 and ψ
would be both zero, which is not a possibility.
Corollary 3. For problem (7), Theorem 3 gives Theorem 2.
Proof. For problem (7) we have ϕ(t, y, u) = u. From Proposition 2, the Hamil-
tonian becomes H(t, y, u, ψ0, ψ
σ) = L(t, y, u) + ψσ · u. By the ∆-stationary
condition (19) we may write Lu(t, y, u) + ψ
σ = 0. Now apply (26) and the
result follows.
To prove Theorem 3 we need the following result:
Lemma 3 (Fundamental lemma of the calculus of variations on time scales).
Let g ∈ Crd, g : T
k → Rn. Then,
g(t) · η(t)∆t = 0 for all η ∈ Crd
if and only if
g(t) = 0 on Tk .
Proof. If g(t) = 0 on Tk, then obviously
g(t) · η(t)∆t = 0, for all η ∈ Crd.
Now, suppose (without loss of generality) that g(t0) > 0 for some t0 ∈ T
We will divide the proof in two steps:
Step 1: Assume that t0 is right scattered. Define in T
η(t) =
1 if t = t0;
0 if t 6= t0.
Then η is rd-continuous and
g(t)η(t)∆t =
∫ σ(t0)
g(t)η(t)∆t = µ(t0)g(t0) > 0,
which is a contradiction.
Step 2: Suppose that t0 is right dense. Since g is rd-continuous, then it is
continuous at t0. So there exist δ > 0 such that for all t ∈ (t0 − δ, t0 + δ) ∩ T
we have g(t) > 0.
If t0 is left-dense, define in T
η(t) =
(t− t0 + δ)
2(t− t0 − δ)
2 if t ∈ (t0 − δ, t0 + δ);
0 otherwise.
It follows that η is rd-continuous and
g(t)η(t)∆t =
∫ t0−δ
g(t)η(t)∆t +
∫ t0+δ
g(t)η(t)∆t +
g(t)η(t)∆t > 0,
which is a contradiction.
If t0 is left-scattered, define in T
η(t) =
(t− t0 − δ)
2 if t ∈ [t0, t0 + δ̃);
0 otherwise,
where 0 < δ̃ < min{µ(ρ(t0), δ)}. We have: η is rd-continuous and
g(t)η(t)∆t =
∫ t0+δ̃
g(t)η(t)∆t > 0,
that again leads us to a contradiction.
Proof. (of Theorem 3) We begin by noting that u(t) = (u1(t), . . . , um(t)) in
problem (16), t ∈ Tk, are arbitrarily specified functions (controls). Once fixed
u(·) ∈ Crd[T;R
m], then y(t) = (y1(t), . . . , yn(t)), t ∈ T
k, is determined from
the system of delta-differential equations y∆(t) = ϕ(t, y(t), u(t)) (and boundary
conditions, if present). As far as u(·) is an arbitrary function, variations ω(·) ∈
Crd[T;R
m] for u(·) can also be considered arbitrary. This is not true, however,
for the variations η(·) ∈C1rd[T;R
n] of y(·). Suppose that (y∗(·), u∗(·)) is a weak
local minimizer of J [·, ·]. Let ε ∈ (−δ, δ) be a small real parameter and yε(t) =
y∗(t)+εη(t) (with η(a) = 0 if y(a) = ya is given; η(b) = 0 if y(b) = yb is given) be
the trajectory generated by the control uε(t) = u∗(t)+εω(t), ω(·) ∈ Crd[T;R
ε (t) = ϕ(t, yε(t), uε(t)) , (27)
t ∈ Tk, (yε(a) = ya), (yε(b) = yb). We define the following function:
Φ(ε) = J [yε(·), uε(·)] = J [y∗(·) + εη(·), u∗(·) + εω(·)]
L (t, y∗(t) + εη(t), u∗(t) + εω(t))∆t .
It follows that Φ : (−δ, δ) → R has a minimum for ε = 0, so we must have
Φ′(0) = 0. From this condition we can write that
[ψ0Ly (t, y∗(t), u∗(t)) · η(t) + ψ0Lu (t, y∗(t), u∗(t)) · ω(t)]∆t = 0 (28)
for any real constant ψ0. Differentiating (27) with respect to ε, we get
η∆(t) = ϕy(t, yε(t), uε(t)) · η(t) + ϕu(t, yε(t), uε(t)) · ω(t) .
In particular, with ε = 0,
η∆(t) = ϕy(t, y∗(t), u∗(t)) · η(t) + ϕu(t, y∗(t), u∗(t)) · ω(t) . (29)
Let ψ(·) ∈C1rd[T;R
n] be (yet) an unspecified function. Multiplying (29) by
ψσ(t) = [ψσ1 (t), . . . , ψ
n(t)], and delta-integrating the result with respect to t
from a to b, we get that
ψσ(t) · η∆(t)∆t =
[ψσ(t) · ϕy · η(t) + ψ
σ(t) · ϕu · ω(t)]∆t (30)
for any ψ(·) ∈ C1rd[T;R
n]. Integrating by parts (see Lemma 1, formula 1),
σ(t) · η∆(t)∆t = ψ(t) · η(t)|
∆(t) · η(t)∆t , (31)
and we can write from (28), (30) and (31) that
ψ∆(t) + ψ0Ly + ψ
σ(t) · ϕy
· η(t)
+ (ψ0Lu + ψ
σ(t) · ϕu) · ω(t)
∆t− ψ(t) · η(t)|
= 0 (32)
hold for any ψ(t). Using the definition (20) of H , we can rewrite (32) as
ψ∆(t) +Hy
· η(t) +Hu · ω(t)
∆t− ψ(t) · η(t)|
= 0 . (33)
It is, however, not possible to employ (yet) Lemma 3 due to the fact that
the variations η(t) are not arbitrary. Now choose ψ(t) = ψ∗(t) so that the
coefficient of η(t) in (33) vanishes: ψ∆
(t) = −Hy (and ψ∗(a) = 0 if y(a) is
free, i.e. η(a) 6= 0; ψ∗(b) = 0 if y(b) is free, i.e. η(b) 6= 0). In the normal case
ψ∗(t) is determined by (y∗(·), u∗(·)), and we choose ψ0∗ = 1. The abnormal case
is characterized by the existence of a non-trivial solution ψ∗(t) for the system
(t) + ψσ
(t) · ϕy = 0: in that case we choose ψ0∗ = 0 in order to the first
coefficient of η(t) in (32) or (33) to vanish. Given this choice of the multipliers,
the necessary optimality condition (33) takes the form
Hu · ω(t)∆t = 0 .
Since ω(t) can be arbitrarily assigned for all t ∈ Tk, it follows from Lemma 3
that Hu = 0.
3.3 The higher-order problem on time scales
As a corollary of Theorem 3 we obtain the Euler-Lagrange equation for problem
(3). We first introduce some notation:
y0(t) = y(t),
y1(t) = y∆(t),
yr−1(t) = y∆
u(t) = y∆
Theorem 4. If y∗ ∈ C
rd[T] is a weak local minimizer for the higher-order
problem (3), then
(σ(t)) = −Lu(t, x∗(t), u∗(t)) (34)
holds for all t ∈ Tk
, where x∗(t) =
y∗(t), y
(t), . . . , y∆
and ψr−1
(σ(t))
is defined recursively by
(σ(t)) = −
∫ σ(t)
Ly0(ξ, x∗(ξ), u∗(ξ))∆ξ + c0 , (35)
(σ(t)) = −
∫ σ(t)
Lyi(ξ, x∗(ξ), u∗(ξ)) + ψ
(σ(ξ))
∆ξ + ci, i = 1, . . . , r − 1 ,
with cj, j = 0, . . . , r − 1, constants. If y
∆i(α) is free in (3) for some i ∈
{0, . . . , r − 1}, α ∈ {a, ρr−1(b)}, then the correspondent condition ψi
(α) = 0
holds.
Remark 12. From (34), (35) and (36) it follows that
(−1)r−i
· · ·
Lyi + [ci]r−i−1 = 0, (37)
where [ci]r−i−1 means that the constant is free from the composition of the r− i
integrals when i = r − 1 (for simplicity, we have omitted the arguments in Lu
and Lyi).
Remark 13. If we delta differentiate (37) r times, we obtain the delta differen-
tiated equation for the problem of the calculus of variations with higher order
delta derivatives. However, as observed in Remark 3, one can only expand
formula (37) under suitable conditions of delta differentiability of µ(t).
Remark 14. For the particular case with ϕ(t, y, u) = u, equation (8) is (37) with
r = 1.
Proposition 3. The higher-order problem on time scales (3) does not admit ab-
normal extremals, even when the boundary conditions y∆
(a) and y∆
(ρr−1(b)),
i = 0, . . . , r − 1, are all fixed.
Remark 15. We require the time scale T to have at least 2r + 1 points. Let
us consider problem (3) with all the boundary conditions fixed. Due to the
fact that we have r delta derivatives, the boundary conditions y∆
(a) = yia and
(ρr−1(b)) = yib for all i ∈ {0, . . . , r− 1}, imply that we must have at least 2r
points in order to have the problem well defined. If we had only this number of
points, then the time scale could be written as T = {a, σ(a), . . . , σ2r−1(a)} and
r−1(b)
L(t, y(t),y∆(t), . . . , y∆
(t))∆t
i+1(a)
σi(a)
L(t, y(t), y∆(t), . . . , y∆
(t))∆t
L(σi(a), y(σi(a)), y∆(σi(a)), . . . , y∆
(σi(a))),
where we have used the fact that ρr−1(σ2r−1(a)) = σr(a). Now, having in mind
the boundary conditions and the formula
f∆(t) =
f(σ(t))− f(t)
we can conclude that the sum in (38) would be constant for every admissible
function y(·) and we wouldn’t have nothing to minimize.
The following technical result is used in the proof of Proposition 3.
Lemma 4. Suppose that a function f : T → R is such that fσ(t) = 0 for all
t ∈ Tk. Then, f(t) = 0 for all t ∈ T\{a} if a is right-scattered.
Proof. First note that, since fσ(t) = 0, then fσ(t) is delta differentiable, hence
continuous for all t ∈ Tk. Now, if t is right-dense, the result is obvious. Suppose
that t is right-scattered. We will analyze two cases: (i) if t is left-scattered,
then t 6= a and by hypothesis 0 = fσ(ρ(t)) = f(t); (ii) if t is left-dense, then
f(t) = lims→t− f
σ(s) = fσ(t), by the continuity of fσ. The proof is done.
Proof. (of Proposition 3) Suppose that ψ0 = 0. With the notation (40) intro-
duced below, the higher order problem (3) would have the abnormal Hamilto-
nian given by
H(t, y0, . . . , yr−1, u, ψ0, . . . , ψr−1) =
ψi(σ(t)) · yi+1(t) + ψr−1(σ(t)) · u(t)
(compare with the normal Hamiltonian (41)). From Theorem 3, we can write
the system of equations:
ψ̂0(t) = 0
ψ̂1(t) = −ψ0(σ(t))
ψ̂r−1(t) = −ψr−2(σ(t))
ψr−1(σ(t)) = 0,
for all t ∈ Tk
, where we are using the notation ψ̂i(t) = ψi
(t), i = 0, . . . , r− 1.
From the last equation, and in view of Lemma 4, we have ψ(t) = 0, ∀t ∈
\{a} if a is right-scattered. This implies that ψ̂r−1(t) = 0, ∀t ∈ Tk
and consequently ψr−2(σ(t)) = 0, ∀t ∈ Tk
\{a}. Like we did before, ψr−2(t) =
0, ∀t ∈ Tk
\{a, σ(a)} if σ(a) is right-scattered. Repeating this procedure, we
will finally have ψ̂1(t) = 0, ∀t ∈ Tk
\{a, . . . , σr−2(a)} if σi(a) is right-scattered
for all i ∈ {0, . . . , r− 2}. Now, the first and second equations in the system (39)
imply that ∀t ∈ A = Tk
\{a, . . . , σr−2(a)}
0 = ψ̂1(t) = −ψ0(σ(t)) = ψ0(t) + µ(t)ψ∆(t) = ψ0(t) .
We pick again the first equation to point out that ψ0(t) = c, ∀t ∈ Tk
some constant c. Since the time scale has at least 2r + 1 points (Remark 15),
the set A is nonempty and therefore ψ0(t) = 0, ∀t ∈ Tk
. Substituting this
in the second equation, we get ψ̂1(t) = 0, ∀t ∈ Tk
. As before, it follows that
ψ1(t) = d, ∀t ∈ Tk
and some constant d. But we have seen that there exists
some t0 such that ψ
1(t0) = 0, hence ψ
1(t) = 0, ∀t ∈ Tk
. Repeating this
procedure, we conclude that for all i ∈ {0, . . . , r−1}, ψi(t) = 0 at t ∈ Tk
. This
is in contradiction with Theorem 3 and we conclude that ψ0 6= 0.
Proof. (of Theorem 4) Denoting ŷ(t) = y∆(t), then problem (3) takes the fol-
lowing form:
L[y(·)] =
r−1(b)
L(t, y0(t), y1(t), . . . , yr−1(t), u(t))∆t −→ min,
ŷ0 = y1
ŷ1 = y2
ŷr−2 = yr−1
ŷr−1 = u
yi(a) = yia
ρr−1(b)
= yib
, i = 0, . . . , r − 1, yia and y
b ∈ R
System (40) can be written in the form y∆ = Ay +Bu, where
y0, y1, . . . , yr−1
y01 , . . . , y
1 , . . . , y
n, . . . , y
∈ Rnr
and the matrices A (nr by nr) and B (nr by n) are
0 I 0 · · · 0
0 0 I · · · 0
. . .
0 0 0 · · · I
0 0 0 · · · 0
, B = col{0, . . . , 0, I}
in which I denotes the n by n identity matrix, and 0 the n by n zero matrix.
From Proposition 3 we can fix ψ0 = 1: problem (40) is a particular case of (16)
with the Hamiltonian given by
H(t, y0, . . . , yr−1, u, ψ0, . . . , ψr−1)
= L(t, y0, . . . , yr−1, u) +
ψi(σ(·)) · yi+1 + ψr−1(σ(·)) · u. (41)
From (26) and (19), we obtain
ψi(σ(t)) = −
∫ σ(t)
Hyi(ξ, x(ξ), u(ξ), ψ
σ(ξ))∆ξ + ci, i ∈ {0, . . . , r − 1} (42)
0 = Hu(t, x(t), u(t), ψ
σ(t)), (43)
respectively. Equation (43) is equivalent to (34), and from (42) we get (35)-
(36).
4 An example
We end with an application of our higher-order Euler-Lagrange equation (37)
to the time scale T = [a, b] ∩ Z, that leads us to the usual and well-known
discrete-time Euler-Lagrange equation (in delta differentiated form) – see e.g.
[11]. Note that ∀t ∈ T we have σ(t) = t + 1 and µ(t) = σ(t) − t = 1. In
particular, we conclude immediately that µ(t) is r times delta differentiable.
Also for any function g, g∆ exists ∀t ∈ Tk (see Theorem 1.16 (ii) of [5]) and
g∆(t) = g(t+1)− g(t) = ∆g is the usual forward difference operator (obviously
exists ∀t ∈ Tk
and more generally g∆
exists ∀t ∈ Tk
, r ∈ N).
Now, for any function f : T → R and for any j ∈ N we have
∫ σ(t)
· · ·
︸ ︷︷ ︸
j−i integrals
, i ∈ {0, . . . , j − 1} , (44)
where f∆
(t) stands for f∆
(σj−i(t)). To see this we proceed by induction.
For j = 1
∫ σ(t)
f(ξ)∆ξ =
∫ t+1
f(ξ)∆ξ =
f(ξ)∆ξ +
∫ t+1
f(ξ)∆ξ
f(ξ)∆ξ + f(t),
and then
∫ σ(t)
f(ξ)∆ξ
= f(t) + f∆(t) = fσ. Assuming that (44) is true for
all j = 1, . . . , k, then
∫ σ(t)
· · ·
︸ ︷︷ ︸
k+1−i integrals
· · ·
︸ ︷︷ ︸
k+1−i
f∆τ +
∫ σ(t)
· · ·
︸ ︷︷ ︸
∫ σ(t)
· · ·
︸ ︷︷ ︸
∫ σ(t)
· · ·
︸ ︷︷ ︸
∆iσk−i
k+1−i
Delta differentiating r times both sides of equation (37) and in view of (44),
we obtain the Euler-Lagrange equation in delta differentiated form (remember
that y0 = y, . . ., yr−1 = y∆
= u):
r (t, y, y∆, . . . , y∆
(−1)r−iL∆
i (t, y, y
, . . . , y
∆r) = 0.
5 Conclusion
We introduce a new perspective to the calculus of variations on time scales. In all
the previous works [2, 4, 9] on the subject, it is not mentioned the motivation for
having yσ (or yρ) in the formulation of problem (1). We claim the formulation
(2) without σ (or ρ) to be more natural and convenient. One advantage of
the approach we are promoting is that it becomes clear how to generalize the
simplest functional of the calculus of variations on time scales to problems with
higher-order delta derivatives. We also note that the Euler-Lagrange equation
in ∆-integral form (8), for a Lagrangian L with y instead of yσ, follows close
the classical condition. Main results of the paper include: necessary optimality
conditions for the Lagrange problem of the calculus of variations on time scales,
covering both normal and abnormal minimizers; necessary optimality conditions
for problems with higher-order delta derivatives. Much remains to be done in
the calculus of variations and optimal control on time scales. We trust that our
perspective provides interesting insights and opens new possibilities for further
investigations.
Acknowledgments
This work was partially supported by the Portuguese Foundation for Science
and Technology (FCT), through the Control Theory Group (cotg) of the Centre
for Research on Optimization and Control (CEOC – http://ceoc.mat.ua.pt).
The authors are grateful to M. Bohner and S. Hilger for useful and stimulating
comments, and for them to have shared their expertise on time scales.
References
[1] R. Agarwal, M. Bohner, D. O’Regan, A. Peterson. Dynamic equations on time
scales: a survey, J. Comput. Appl. Math. 141 (2002), no. 1-2, 1–26.
[2] F. M. Atici, D. C. Biles, A. Lebedinsky. An application of time scales to economics,
Math. Comput. Modelling 43 (2006), no. 7-8, 718–726.
[3] Z. Bartosiewicz, E. Paw luszewicz. Realizations of linear control systems on time
scales, Control Cybernet. 35 (2006), no. 4 (in press)
[4] M. Bohner. Calculus of variations on time scales, Dynam. Systems Appl. 13
(2004), no. 3-4, 339–349.
[5] M. Bohner, A. C. Peterson. Dynamic equations on time scales: an introduction
with applications, Birkhäuser Boston, Inc., Boston, MA, 2001.
[6] I. M. Gelfand, S. V. Fomin. Calculus of variations, Dover, New York, 2000.
[7] S. Hilger. Analysis on measure chains—a unified approach to continuous and dis-
crete calculus, Results Math. 35 (1990), 18–56.
[8] S. Hilger. Differential and difference calculus—unified!, Proceedings of the Second
World Congress of Nonlinear Analysts, Part 5 (Athens, 1996). Nonlinear Anal.
30 (1997), no. 5, 2683–2694.
[9] R. Hilscher, V. Zeidan. Calculus of variations on time scales: weak local piecewise
rd solutions with variable endpoints, J. Math. Anal. Appl. 289 (2004), no. 1,
143–166.
[10] R. Hilscher, V. Zeidan. Nonnegativity and positivity of quadratic functionals in
discrete calculus of variations: survey, J. Difference Equ. Appl. 11 (2005), no. 9,
857–875.
[11] J. D. Logan. Higher dimensional problems in the discrete calculus of variations,
Internat. J. Control (1) 17 (1973), 315–320.
[12] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko. The
mathematical theory of optimal processes. Translated from the Russian by K. N.
Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons,
Inc. New York-London, 1962.
[13] S. P. Sethi, G. L. Thompson. Optimal control theory, Second edition, Kluwer
Acad. Publ., Boston, MA, 2000.
[14] B. van Brunt. The calculus of variations, Universitext, Springer-Verlag, New York,
2004.
	Introduction
	Time scales and previous results
	Main results
	The basic problem on time scales
	The Lagrange problem on time scales
	The higher-order problem on time scales
	An example
	Conclusion
