doi:10.1155/2009/840386
Research Article
Vartiational Optimal-Control Problems with
Delayed Arguments on Time Scales
Thabet Abdeljawad (Maraaba),
1Fahd Jarad,
1and Dumitru Baleanu
1, 21Department of Mathematics and Computer Science, C¸ankaya University, 06530 Ankara, Turkey 2Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele-Bucharest, Romania
Correspondence should be addressed to Thabet AbdeljawadMaraaba,[email protected]
Received 11 August 2009; Revised 3 November 2009; Accepted 16 November 2009
Recommended by Paul Eloe
This paper deals with variational optimal-control problems on time scales in the presence of delay in the state variables. The problem is considered on a time scale unifying the discrete, the continuous, and the quantum cases. Two examples in the discrete and quantum cases are analyzed to illustrate our results.
Copyrightq2009 Thabet AbdeljawadMaraabaet al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The calculus of variations interacts deeply with some branches of sciences and engineering, for example, geometry, economics, electrical engineering, and so on 1. Optimal control problems appear in various disciplines of sciences and engineering as well2.
Time-scale calculus was initiated by Hilgersee3and the references thereinbeing in mind to unify two existing approaches of dynamic models difference and differential equations into a general framework. This kind of calculus can be used to model dynamic processes whose time domains are more complex than the set of integers or real numbers
4. Several potential applications for this new theory were reportedsee, e.g.,4–6and the references therein. Many researchers studied calculus of variations on time scales. Some of them followed the delta approach and some others followed the nabla approachsee, e.g.,
7–12.
knowledge, there is no work in the direction of variational optimal-control problems with delayed arguments on time scales.
Our aim in this paper is to obtain the Euler-Lagrange equations for a functional, where the state variables of its Lagrangian are defined on a time scale whose backward jumping operator is ρt qt−h,q > 0,h ≥ 0. This time scale, of course, absorbs the discrete, the continuous and the quantum cases. The state variables of this Lagrangian allow the presence of delay as well. Then, we generalize the results to then-dimensional case. Dealing with such a very general problem enables us to recover many previously obtained results14–17.
The structure of the paper is as follows. InSection 2basic definitions and preliminary concepts about time scale are presented. The nabla time-scale derivative analysis is followed there. In Section 3 the Euler-Lagrange equations into one unknown function and then in then-dimensional case are obtained. InSection 4the variational optimal control problem is proposed and solved. InSection 5the results obtained in the previous sections are particulary studied in the discrete and quantum cases, where two examples are analyzed in details. Finally,Section 6contains our conclusions.
2. Preliminaries
A time scale is an arbitrary closed subset of the real lineR. Thus the real numbers and the natural numbers,N, are examples of a time scale. Throughout this paper, and following4, the time scale will be denoted byT. The forward jump operatorσ:T → Tis defined by
σt:inf{s∈T:s > t}, 2.1
while the backward jump operatorρ:T → Tis defined by
ρt:sup{s∈T:s < t}, 2.2
where, inf∅ supTi.e.,σt tifThas a maximumtand sup∅ infTi.e.,ρt tifT has a minimumt. A pointt∈Tis called right-scattered ift < σt, left-scattered ifρt< t, and isolated if ρt < t < σt. In connection we define the backward graininess function ν:T → 0,∞by
νt t−ρt. 2.3
In order to define the backward time-scale derivative down, we need the set Tκ which is derived from the time scaleTas follows. IfThas a right-scattered minimumm, thenTκ
T− {m}. Otherwise,TκT.
Definition 2.1see18. Assume thatf:T → Ris a function andt∈Tκ. Then the backward time-scale derivativef∇tis the numberprovided that it existswith the property that given any >0,there exists a neighborhoodUofti.e.,U t−δ, tδfor someδ >0such that
fs−f
ρt−s−ρt≤s−ρt ∀s∈U. 2.4
The following theorem is Theorem 3.2 in19and an analogue to Theorem 1.16 in4.
Theorem 2.2see 18. Assume thatf : T → R is a function andt ∈ Tκ, then one has the
following.
iIffis differentiable attthenfis continuous att.
iiIffis continuous attandtis left-scattered, thenfis differentiable attwith
f∇t ft−f
ρt
νt . 2.5
iiiIftis left-dense, then f is differentiable attif and only if the limit
lim s→t
ft−fs
t−s 2.6
exists as a finite number. In this case
f∇t lim s→t
ft−fs
t−s . 2.7
ivIffis∇-differentiable att, then
ft fρtνtf∇t. 2.8
Example 2.3. iT Ror any any closed interval the continuous case σt ρt t, νt 0,andf∇t ft.
iiThZ,h >0 or any subset of itthe difference calculus, a discrete caseσt th, ρt t−h,νt h, andf∇t ∇hft ft−ft−h.
iiiTTq {qn :n∈Z} ∪ {0}, 0< q <1,quantum calculusσt q−1t,ρt qt, νt 1−qt, andf∇t ∇qft ft−fqt/1−qt.
ivTTh
q {qk− k−2
i0 qih:k ≥2, k∈N} ∪ {−h/1−q}, 0< q <1,h >0unifying the difference calculus and quantum calculus. There areσt q−1th, ρt qt−h,
νt 1−qth, andf∇t ∇h
qft ft−fqt−h/1−qth. Ifα0∈Nthenρα0t
qα0t−α0k0−1qkhand so∇hqρα0t qα0. Note that in this example the backward operator is of the formρt ctdand henceTh
q is an element of the classHof time scales that contains the discrete, the usual, and the quantum calculussee17.
Theorem 2.4. Suppose thatf, g:T → Rare nabla differentiable att∈Tκ, then,
1the sumfg:T → Ris nabla differentiable attandfg∇t f∇t g∇t;
2for anyλ∈R, the functionλf :T → Ris nabla differentiable attandλf∇t λf∇t;
3the productfg :T → Ris nabla differentiable attand
For the proof of the following lemma we refer to20.
Lemma 2.5. Let T be an H-time scale (in particular T Th
q), f : T → R two times nabla
differentiable function, andgt ρα0t,forα
0∈N. Then
f◦g∇t f∇gt·g∇t, t∈Tκ. 2.10
Throughout this paper we use for the time-scale derivatives and integrals the symbol ∇h
q which is inherited from the time scaleThq. However, our results are true also for the H-time scalesthose time scales whose jumping operators have the formatb. The time scale Th
q is a natural example of anH-time scale.
Definition 2.6. A functionF:T → Ris called a nabla antiderivative off :T → Rprovided Ft ft,for allt∈Tκ. In this case, fora, b∈T, we write
b
a
ft∇t:Fb−Fa. 2.11
The following lemma which extends the fundamental lemma of variational analysis on time scales with nabla derivative is crucial in proving the main results.
Lemma 2.7. Letg ∈Cld,g:a, b → Rn. Then
b
a
gTtη∇t∇t ∀η∈C1ld withηa ηb 0 2.12
holds if and only if
gt≡c ona, bκ for somec∈Rn. 2.13
The proof can be achieved by following as in the proof of Lemma 4.1 in9 see also
17.
3. First-Order Euler-Lagrange Equation with Delay
We consider theTh
q-integral functionalJ:S → R,
Jy b
a
where
a, b∈Thq, a < ρα0b< b,
L:a, b×Rn4 −→R, yρx yρx,
Sy:ρα0a, b−→Rn:yx ϕx∀x∈ρα0a, a, yb c0
.
3.2
We will shortly write
Lx≡L x, yρx,∇hqyx, yρρα0x,∇hqyρα0x. 3.3
We calculate the first variation of the functionalJ on the linear manifoldS. Letη ∈ H{h:ρα0a, b → Rn:hx 0∀x∈ρα0a, a∪ {b}}, then
δJyx, ηx d dJ
yx ηx 0,
b
a
∂1Lxηρx ∂2Lx∇hqηx ∂3Lxηρ
ρα0xqα0∂4Lx∇hqη
ρα0x∇hqx,
3.4
where
∂1L
∂L
∂yρx, ∂2L
∂L
∂ ∇h qyx
, ∂3L
∂L
∂yρρα0x, ∂4L
∂L
∂ ∇h qy
ρα0x,
3.5
and where Lemma 2.5and that∇h
qρα0t qα0 are used. If we use the change of variable u ρα0x, which is a linear function, and make use of Theorem 1.98 in4andLemma 2.5 we then obtain
δJyx, ηx b
a
∂1Lxηρx ∂2Lx∇hqηx
∇h qx
ρα0b
a
q−α0∂3L ρα0
−1
xηρx q−α0∂4L ρα0
−1
x∇hqηx∇hqx,
3.6
Then the necessary condition forJyto possess an extremum for a given functionyxis thatyx
satisfies the following Euler-Lagrange equations
∇h
The necessary condition represented by3.12is obtained by applying integration by parts in3.7and then substituting3.11in the resulting integrals. The above theorem can be generalized as follows.
Then a necessary condition for Jy to possess an extremum for a given function yx
Furthermore, the equations
q−α0∂4Li ρα0
−1
xηix ρα0b
a 0 3.16
hold alongyxfor all admissible variationsηixsatisfying
ηix 0, x∈
ρα0a, a∪ {b}, i1,2, . . . , m, 3.17
where
∂1Li
∂L
∂ yiρx
, ∂2Li
∂L
∂ ∇hqyix
, ∂3Li
∂L
∂ yρiρα0x, ∂4L
i ∂L
∂ ∇hqyi
ρα0x.
3.18
4. The Optimal-Control Problem
Our aim in this section is to find the optimal control variable ux defined on theH-time scale, which minimizes the performance index
Jy, u b
a
L x, yρx, uρx, yρρα0x,∇hqyρα0x∇hqx 4.1
subject to the constraint
∇h
qyx G
x, yρx, uρx 4.2
such that
yb c, yx φx x∈ρα0a, a,
a, b∈Thq, a < ρα0b< b,
L:a, b×Rn4 −→R, yρx yρx,
4.3
where cis a constant and L and Gare functions with continuous first and second partial derivatives with respect to all of their arguments. To find the optimal control, we define a modified performance index as
Iy, u b
a
L x, yρx, uρx, yρρα0x,∇hqyρα0x
λρx ∇hqyx−Gx, yρx, uρx∇hqx,
4.4
Using 3.11 and 3.12 of Theorem 3.2with m 3 y1 y,y2 u,y3 λ, the
necessary conditions for our optimal control arewe remark that as there is no any time-scale derivative ofux, no boundary constraints for it are needed
∇h
Note that condition4.6disappears when the LagrangianLis free of the delayed time scale derivative ofy.
5. The Discrete and Quantum Cases
We recall that the results in the previous sections are valid for time scales whose backward jump operatorρhas the formρx qx−h, in particular for the time scaleTh
q.
(i) The Discrete Case
The necessary condition forJhyto possess an extremum for a given functiony:{ih: ia−d, a−d1, . . . , a, a1, . . . , b} → Rnis thatyxsatisfies the followingh-Euler-Lagrange equations:
∇h∂
2Lih ∇h∂4Lidh ∂1Lih ∂3Lidh ia1, a2, . . . , b−d,
∇h∂
2Lih ∂1Lih ib−d1, b−d2, . . . , b.
5.3
Furthermore, the equation
∂4Lbhηb−dh−∂4Ladhηah 0 5.4
holds alongyxfor all admissible variationsηxsatisfyingηih 0,i ∈ {a−d, a−d 1, . . . , a} ∪ {b}.
In this case theh-optimal-control problem would read as follows.
Find the optimal control variableuxdefined on the time scalehZ, which minimizes theh-performance index
Jhy, uh b
ia1
L ih, yi−1h, ui−1h, yih−d1h,∇hyih−dh,
a, b∈Z, d∈N, a < b−d < b,
5.5
subject to the constraint
∇hyih Gih, yi−1h, ui−1h, ia1, a2, . . . , b, 5.6
such that
ybh c, yih φih ia−d, a−d1, . . . , a,
The necessary conditions for thish-optimal control are
∇hλi−1h ∇h ∂L
∂∇hyi−d−1hidh λi−1h ∂G ∂yi−1h − ∂L
∂yi−1h−
∂L
∂yi−d−1hidh 0 ia1, a2, . . . , b−d,
∇h
qλi−1h λi−1h ∂G ∂yi−1h−
∂L
∂yi−1h 0 ib−d1, b−d2, . . . , b,
λi−1h ∂G ∂ui−1h −
∂L
∂ui−1h 0 ia, a1, . . . , b,
5.8
∂L
∂∇hyidhbhηb−dh−
∂L
∂∇hyidhadhηah 0, 5.9
and also
∇hyih Gih, yi−1h, ui−1h, ia1, a2, . . . , b. 5.10
Note that condition 5.9 disappears when the Lagrangian L is independent of the delayed∇hderivative ofy.
Example 5.1. In order to illustrate our results we analyze an example of physical interest. Namely, let us consider the following discrete action:
Jht h 2
b
ia1
∇hyih2−Vyih−d1h, a, b∈N, a < b−d < b 5.11
subject to the condition
ybh c, yih ϕih, foria−d, a−d1, . . . , a. 5.12
The correspondingh-Euler-Lagrange equations are as follows:
yih−2yi−1h yi−2h ∂V
∂yih−d1hidh ia1, . . . , b−d,
yih−2yi−1h yi−2h 0 ib−d1, b−d2, . . . , b.
5.13
(ii) The Quantum Case
If 0< q < 1 andh 0, then our work becomes on the time scaleTq {qn :n∈Z} ∪ {0}. In this case the functional under optimization will have the form
Jq
Using the∇-integral theory on time scales, the functionalJqin5.14turns to be
Jq
The necessary condition forJqyto possess an extremum for a given function y : {qi : i α1−α In this case theq-optimal-control problem would read as follows.
subject to the constraint
where cis a constant and L and Gare functions with continuous first and second partial derivatives with respect to all of their arguments.
The necessary conditions for thisq-optimal control are
∇qλ
Note that condition5.25disappears when the LagrangianL is independent of the delayed∇qderivative ofy.
Example 5.2. Suppose that the problem is that of finding a control functionuxdefined on the time scaleTqsuch that the corresponding solution of the controlled system
∇qyx −ry
qxuqx, r >0, 5.27
satisfying the conditions
is an extremum for theq-integral functionalq-quadratic delay cost functional:
Jq
yx, ux 1 2
1−q β
iα
qiy2 qiα01u2 qi1. 5.29
According to5.24and5.25, the solution of the problem satisfies
∇qλ
qxrλqxq−α0yqx x∈a, qα0bκ, ∇qλ
qxrλqx x∈qα0b, bκ,
λqxuqx x∈a, bq,
5.30
and of course
∇qyx −ry
qxuqx. 5.31
When the delay is absenti.e.,α0 0, it can be shown that the above system is reduced
to a second-orderq-difference equation. Namely, reduced to
∇2
qyx rq
∇qy
qxq r21yqxqr∇qyx. 5.32
If we solve recursively for this equation in terms of an integer power series by using the initial data, then the resulting solution will tend to the solutions of the second order linear differential equation:
y− r21y0. 5.33
Clearly the solutions for this equation are exp√r21xand exp−√r21x. For details see
16.
6. Conclusion
absent some of the results in15are reobtained. When the delay is absent and the time scale is free somehow, some of the results in17can be recovered as well.
Finally, we would like to mention that we followed the line of nabla time-scale derivatives in this paper, analogous results can be originated if the delta time-scale derivative approach is followed.
Acknowledgment
This work is partially supported by the Scientific and Technical Research Council of Turkey.
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