Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales

0.13 MB·English

by  Agnieszka B. Malinowska

Leitmann’s direct method of optimization
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for absolute extrema of certain problems
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0
2 of the calculus of variations on time scales∗

n
a
J
Agnieszka B. Malinowska† Delfim F. M. Torres
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[email protected] [email protected]

]
C
O Department of Mathematics
h. University of Aveiro
t 3810-193 Aveiro, Portugal
a
m
[

1
v Abstract
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The fundamental problem of the calculus of variations on time
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1 scalesconcernstheminimizationofadelta-integraloveralltrajectories
. satisfying given boundary conditions. This includes the discrete-time,
1
0 the quantum, and the continuous/classical calculus of variations as
0
particular cases. In this note we follow Leitmann’s direct method to
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: give explicit solutions for some concrete optimal control problems on
v
an arbitrary time scale.
i
X
r
a Mathematics Subject Classification 2000: 49J05; 49M30; 39A12.
Keywords: calculusofvariations,optimalcontrol, Leitmann’sdirectmethod,
time scales.
∗Dedicated to George Leitmann on the occasion of his 85th birthday. The authors
are grateful to George Leitmann for many helpful remarks and discussions. Accepted for
publication (9/January/2010)in Applied Mathematics and Computation.
†On leave of absence from Bial ystok University of Technology, Poland (abma-
[email protected]).
1
1 Introduction
The calculus on time scales is a recent field that unifies the theories of differ-
ence and differential equations. It has found applications in several contexts
that require simultaneous modeling of discrete and continuous data, and is
nowadays under strong current research in several different areas [6, 7].
The area of the calculus of variations on time scales, which we are con-
cerned in this paper, was born in 2004 [5] and is now receiving a lot of
attention, both for theoretical and practical reasons – see [1, 2, 3, 4, 12,
13, 21, 22, 23] and references therein. Although the theory is already well
developed in many directions, a crucial problem still persists: solving Euler-
Lagrange delta-differential equations on arbitrary time scales is difficult or
even impossible. As a consequence, there is a lack of concrete variational
problems for which a solution is known. In this paper we follow a different
approach. We show that the direct method introduced by Leitmann in the
sixties of the XX century [15] can also be applied to variational problems on
time scales. Leitmann’s method is a venerable forty years old method that
has shown through the times to be an universal and useful method in several
different contexts – see, e.g., [8, 9, 10, 11, 16, 17, 18, 19, 20, 24, 25, 26]. Here
we provide concrete examples of problems of the calculus of variations on
time scales for which a global minimizer is easily found by the application of
Leitmann’s direct approach.
2 Preliminaries on time scales
A time scale T is an arbitrary nonempty closed subset of the set R of real
R
numbers. It is a model of time. Besides standard cases of (continuous
Z
time) and (discrete time), many different models are used. For each time
T
scale the following operators are used:
• the forward jump operator σ : T→T, σ(t) := inf{s ∈ T : s > t} for
t < supT and σ(supT) = supT if supT < +∞;
• the backward jump operator ρ : T→T, ρ(t) := sup{s ∈ T : s < t} for
t > infT and ρ(infT) = infT if infT > −∞;
• the graininess function µ : T→[0,∞), µ(t) := σ(t)−t.
2
For T = R one has σ(t) = t = ρ(t) and µ(t) ≡ 0 for any t ∈ R. For T = Z
one has σ(t) = t+1, ρ(t) = t−1, and µ(t) ≡ 1 for every t ∈ Z.
A point t ∈ T is called: (i) right-scattered if σ(t) > t, (ii) right-dense if
σ(t) = t, (iii) left-scattered if ρ(t) < t, (iv) left-dense if ρ(t) = t, (v) isolated if
it is both left-scattered and right-scattered, (vi) dense if it is both left-dense
and right-dense. If supT is finite and left-scattered we set Tκ := T\{supT};
otherwise, Tκ := T.
T
We assume that a time scale has the topology that it inherits from the
real numbers with the standard topology. Let f : T→R and t ∈ Tκ. The
delta derivative of f at t is the real number f∆(t) with the property that
given any ε there is a neighborhood U = (t−δ,t+δ)∩T of t such that
∆
|(f(σ(t))−f(s))−f (t)(σ(t)−s)| ≤ ε|σ(t)−s|
for all s ∈ U. We say that f is delta-differentiable on T provided f∆(t) exists
for all t ∈ Tκ.
We note that if T = R, then f : R→R is delta differentiable at t ∈ R if
and only if f is differentiable in the ordinary sense at t. Then, f∆(t) = f′(t).
If T = Z, then f : Z→R is always delta differentiable at every t ∈ Z with
f∆(t) = f(t+1)−f(t).
A function f : T → R is called rd-continuous if it is continuous at the
T
right-dense points in and its left-sided limits exist at all left-dense points
in T. The set of all rd-continuous functions is denoted by C . Similarly, C1
rd rd
will denote the set of functions from C whose delta derivative belongs to
rd
C . A continuous function f is piecewise rd-continuously delta-differentiable
rd
(we write f ∈ C1 ) if f is continuous and f∆ exists for all, except possibly at
prd
finitely many t ∈ Tκ, and f∆ ∈ C . It is known that piecewise rd-continuous
rd
functions possess an antiderivative, i.e., there exists a function F with F∆ =
d
f, and in this case the delta-integral is defined by f(t)∆t = F(d)−F(c)
c
for all c,d ∈ T. If T = R, then
R
b b
f(t)∆t = f(t)dt,
Z Z
a a
where the integral on the right hand side is the usual Riemann integral; if
T = hZ, h > 0, and a < b, then
b b−1
h
f(t)∆t = h·f(kh).
Za kX=ha
3
For f : T→X, where X is an arbitrary set, we define fσ := f ◦σ.
3 Leitmann’s direct method on time scales
Let T be a time scale with at least two points. Throughout we let a,b ∈ T
with a < b. For an interval [a,b]∩T we simply write [a,b].
The problem of the calculus of variations on time scales consists in mini-
mizing
b
L[x] = L(t,xσ(t),x∆(t))∆t
Za
over all x ∈ C1 ([a,b],R) satisfying the boundary conditions
prd
x(a) = α, x(b) = β, (1)
where α, β ∈ R and L : [a,b]κ × R × R → R. We assume that (t,y,v) →
L(t,y,v) has partial continuous derivatives L and L , respectively with re-
y v
spect to the second and third arguments, for all t ∈ [a,b]κ, and L(·,y,v),
L (·,y,v)andL (·,y,v)arepiecewiserd-continuousintforallx ∈ C1 ([a,b],R).
y v prd
A function x ∈ C1 ([a,b],R) is said to be admissible if it satisfies the bound-
prd
ary conditions (1).
Let L˜ : [a,b]κ × R × R → R. We assume that (t,y,v) → L˜(t,y,v) has
partial continuous derivatives L˜ and L˜ , respectively with respect to the
y v
second and third arguments, for all t ∈ [a,b]κ, and L˜(·,y,v), L˜ (·,y,v) and
y
L˜ (·,y,v) are piecewise rd-continuous in t for all x ∈ C1 ([a,b],R). Consider
v prd
the integral
b
L˜[x˜] = L˜(t,x˜σ(t),x˜∆(t))∆t
Za
Lemma 3.1 (Leitmann’s fundamental lemma). Let x = z(t,x˜) be a trans-
formation having an unique inverse x˜ = z˜(t,x) for all t ∈ [a,b] such that
there is a one-to-one correspondence
x(t) ⇔ x˜(t),
for all functions x ∈ C1 ([a,b],R) satisfying (1) and all functions x˜ ∈
prd
C1 ([a,b],R) satisfying
prd
x˜ = z˜(a,α), x˜ = z˜(b,β). (2)
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If the transformation x = z(t,x˜) is such that there exists a function G :
[a,b]×R → R satisfying the functional identity
L(t,xσ(t),x∆(t))−L˜(t,x˜σ(t),x˜∆(t)) = G∆(t,x˜(t)), (3)
then if x˜∗ yields the extremum of L˜ with x˜∗ satisfying (2), x∗ = z(t,x˜∗) yields
the extremum of L for x∗ satisfying (1).
Proof. The proof is similar in spirit to Leitmann’s proof [15, 16, 17, 20]. Let
x ∈ C1 ([a,b],R) satisfy (1) and define functions x˜ ∈ C1 ([a,b],R) through
prd prd
the formula x˜ = z˜(t,x), a ≤ t ≤ b. Then x˜ ∈ C1 ([a,b],R) and satisfies (2).
prd
Moreover, as a result of (3), it follows that
b b
L[x]−L˜[x˜] = L(t,xσ(t),x∆(t))∆t− L˜(t,x˜σ(t),x˜∆(t))∆t
Za Za
b
∆
= G (t,x˜(t))∆t = G(b,x˜(b))−G(a,x˜(a))
Za
= G(b,z˜(b,β))−G(a,z˜(a,β)),
from which the desired conclusion follows immediately since the right-hand
side of the above equality is a constant depending only on the fixed-endpoint
conditions (1).
4 An illustrative example
Let a,b ∈ T, a < b, and α and β be two given reals, α 6= β. We consider the
following problem of the calculus of variations on time scales:
b
minimize L[x] = (x∆(t))2 +xσ(t)+tx∆(t) ∆t,
(4)
Za
x(a) =(cid:0)α, x(b) = β. (cid:1)
We transform problem (4) into the trivial problem
b
minimize L˜[x˜] = (x˜∆(t))2∆t,
Za
x˜(a) = 0, x˜(b) = 0,
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which has the solution x˜ ≡ 0. For that we consider the transformation
x(t) = x˜(t)+ct+d, c,d ∈ R,
where constants c and d will be chosen later. According to the above, we
have
x∆(t) = x˜∆(t)+c, xσ(t) = x˜σ(t)+cσ(t)+d
and
L(t,xσ(t),x∆(t)) = (x∆(t))2 +xσ(t)+tx∆(t)
= (x˜∆(t))2 +2cx˜∆(t)+c2 +x˜σ(t)+cσ(t)+d+tx˜∆(t)+ct
= L˜(t,x˜σ(t),x˜∆(t))+[2cx˜(t)+tx˜(t)+ct2 +(c2 +d)t]∆.
In order to obtain the solution to the original problem, it suffices to chose c
and d so that
ca+d = α
(5)
(cb+d = β.
Solving thesystem of equations (5) we obtainc = α−β andd = βa−bα. Hence,
a−b a−b
the global minimizer to problem (4) is
α−β βa−bα
x(t) = t+ .
a−b a−b
5 Optimal control on time scales
The study of more general problems of optimal control on time scales is in
its infancy, and results are rare (see [22, 27] for some preliminary results).
Similar to the calculus of variations on time scales, there is a lack of exam-
ples with known solution. Here we solve an optimal control problem on an
arbitrary time scale using the idea of Leitmann’s direct method. Consider
the global minimum problem
1
2 2
minimize L[u1,u2] = (u1(t)) +u2(t)) ∆t (6)
0
Z
(cid:0) (cid:1)
subject to the control system
x∆1 (t) = exp(u1(t))+u1(t)+u2(t),
(7)
(x∆2 (t) = u2(t),
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and conditions
x1(0) = 0, x1(1) = 2, x2(0) = 0, x2(1) = 1,
(8)
u1(t),u2(t) ∈ Ω = [−1,1].
This example is inspired in [25]. It is worth to mention that a theory based
on necessary optimality conditions on time scales to solve problem (6)-(8)
does not exist at the moment.
Webeginnoticingthatproblem(6)-(8)isvariationallyinvariantaccording
1
to [14] under the one-parameter transformations
xs1 = x1 +st, xs2 = x2 +st, us2 = u2 +s (ts = t and us1 = u1). (9)
To prove this, we need to show that both the functional integral L[·] and
the control system stay invariant under the s-parameter transformations (9).
This is easily seen by direct calculations:
1
Ls[us,us] = (us(t))2 +(us(t))2∆t
1 2 1 2
0
Z
1
2 2
= u1(t) +(u2(t)+s) ∆t
(10)
0
Z
1
2 2 2 ∆
= u1(t) +u2(t) +[s t+2sx2(t)] ∆t
0
Z
= L[u1(cid:0),u2]+s2 +2s. (cid:1)
We remark that Ls and L have the same minimizers: adding a constant
s2+2s to the functional L does not change the minimizer of L. It remains to
prove that the control system also remains invariant under transformations
(9):
(xs1(t))∆ = (x1(t)+st)∆ = x∆1 (t)+s = exp(u1(t))+u1(t)+u2(t)+s
= exp(us(t))+us(t)+us(t), (11)
1 1 2
(xs2(t))∆ = (x2(t)+st)∆ = x∆2 (t)+s = u2(t)+s = us2(t).
Conditions (10) and (11) prove that problem (6)-(8) is invariant under the
s-parameter transformations (9) up to (s2t+2sx2)∆. Using the invariance
1
A computer algebra package that can be used to find the invari-
ance transformations is available from the Maple Application Center at
http://www.maplesoft.com/applications/view.aspx?SID=4805
7
transformations (9), we generalize problem (6)-(8) to a s-parameter family
of problems, s ∈ R, which include the original problem for s = 0:
1
minimize Ls[us,us] = (us(t))2 +(us(t))2∆t
1 2 1 2
0
Z
subject to the control system
∆
(xs(t)) = exp(us(t))+us(t)+us(t),
1 1 1 2
((xs2(t))∆ = us2(t),
and conditions
xs(0) = 0, xs(1) = 2+s, xs(0) = 0, xs(1) = 1+s,
1 1 2 2
us(t) ∈ [−1,1], us(t) ∈ [−1+s,1+s].
1 2
It is clear that Ls ≥ 0 and that Ls = 0 if us(t) = us(t) ≡ 0. The control
1 2
equations, 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:
1 2
xs=−1(t) = t, xs=−1(t) ≡ 0. Hence, for s = −1 the global minimum to Ls is
1 2
0 and the minimizing trajectory is given by
u˜s(t) ≡ 0, u˜s(t) ≡ 0, x˜s(t) = t, x˜s(t) ≡ 0.
1 2 1 2
Since for any s one has by (10) that L[u1,u2] = Ls[us1,us2] − s2 − 2s, we
conclude that the global minimum for problem L[u1,u2] is 1. Thus, using
the inverse functions of the variational symmetries (9),
u1(t) = us1(t), u2(t) = us2(t)−s, x1(t) = xs1(t)−st, x2(t) = xs2(t)−st.
The absolute minimizer for problem (6)-(8) is
u˜1(t) = 0, u˜2(t) = 1, x˜1(t) = 2t, x˜2(t) = t.
Acknowledgements
Malinowska is a post-doc researcher at the University of Aveiro with the
support of Bial ystok University of Technology via a project of the Polish
Ministry of Science and Higher Education. Torres was partially supported
by the R&D unit CEOC, via FCT and the EC fund FEDER/POCI 2010,
and partially supported by the research project UTAustin/MAT/0057/2008.
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