Fractional Noether's theorem in the Riesz-Caputo sense

0.25 MB·English

by  Gastao S. F. Frederico

Fractional Noether’s theorem
0 in the Riesz-Caputo sense∗
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0
2 Gasta˜o S. F. Frederico

n [email protected]
a
Department of Science and Technology
J
University of Cape Verde
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2 Praia, Santiago, Cape Verde

] Delfim F. M. Torres†
C
[email protected]
O
Department of Mathematics
.
h
University of Aveiro
t
a 3810-193 Aveiro, Portugal
m
[

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Abstract
v
7
WeproveaNoether’stheoremforfractionalvariationalproblemswith
0
Riesz-Caputoderivatives. BothLagrangianandHamiltonianformulations
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areobtained. Illustrativeexamplesinthefractionalcontextofthecalculus
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of variations and optimal control are given.
.
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0 Keywords: calculus of variations, optimal control, fractional deriva-
0 tives, invariance, Noether’s theorem, Leitmann’s direct method.
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: 2000 Mathematics Subject Classification: 49K05, 26A33.
v
i
X
r
a
1 Introduction
Variational symmetries are defined by parameter transformations that keep a
problem of the calculus of variations or optimal control invariant [28, 35, 51].
Their importance, as recognized by Noether in 1918, is connected with the
existenceofconservationlawsthatcanbeusedtoreducetheorderoftheEuler-
Lagrange differential equations [21, 27, 50]. Noether’s symmetry theorem is
∗Accepted(25/Jan/2010) forpublicationinApplied Mathematicsand Computation.
†Correspondingauthor. Partiallysupported by the Centre for Research on Optimization
and Control (CEOC) of the University of Aveiro, cofinanced by the European Community
fundFEDER/POCI2010.
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nowadays recognized as one of the most beautiful results of the calculus of
variations and optimal control [14, 49, 52].
In1967adirectmethodfortheproblemsofthecalculusofvariations,which
allowtoobtainabsoluteextremizersdirectly,without usingthe Euler-Lagrange
equations,wasintroducedbyGeorgeLeitmann[34,36,37]. Timeasshownthat
Leitmann’s method is a general and fruitful principle that can be applied with
success to a myriad of different classes of problems [8, 9, 10, 11, 38, 39, 40].
Interestingly, it turns out that Leitmann’s and Noether’s principles are closely
connected [48, 53].
Thefractionalcalculusisanareaofcurrentstrongresearchwithmanydiffer-
ent andimportantapplications [31, 43, 45, 47]. In the last yearsits importance
in the calculus of variations and optimal control has been perceived, and a
fractionalvariationaltheory began to be developed by severaldifferent authors
[1, 7, 12, 18, 19, 29, 44, 46]. Most part of the results in this direction make
useoffractionalderivativesinthesenseofRiemann-Liouville[7,18,23,25,44],
FALVA [19, 20, 22], or Caputo [1, 5, 24]. In 2007 generalized Euler-Lagrange
fractional equations and transversality conditions were studied for variational
problemsdefinedintermsofRieszfractionalderivatives[2]. Inthispaperwede-
velopfurtherthetheorybyobtainingafractionalversionofNoether’ssymmetry
theorem for variational problems with Riesz-Caputo derivatives (Theorems 24
and 35). Both fractional problems of the calculus of variations and optimal
control are considered.
We finish this introduction comparing in some details the results here ob-
tained with the ones of references [18, 19, 20].
In[18]a(α,β)fractionalderivativeisconsidered,whichinvolvesarightfrac-
tional derivative of order α and a left fractional derivative of order β combined
using a complex number γ. The (α,β) fractional derivative is useful when one
needs to deal with complex valued functions. In our paper we consider real
valued functions only. Moreover, we consider left and right derivatives in the
sense of Caputo, while the (α,β) derivative in [18] is defined via left and right
Riemann-Liouville derivatives. The advantage of using the Riesz symmetrized
Caputo fractional derivative instead of the (α,β) derivative in [18] is that Ca-
putoderivativesallowustousethestandardboundaryconditionsofthecalculus
ofvariations,whichexplainswhytheyaremorepopularinengineerandphysics.
Inour paperwe considerproblemsofthe calculus ofvariationsfor functions
withoneindependentvariable. Paper[19]initiatesanewareaoffractionalvari-
ational calculus by proposing a fractional variational theory involving multiple
integrals. Some importantconsequences of such theory in mechanicalproblems
involving dissipative systems with infinitely many degrees of freedom are given
in[19],butaformaltheoryforthatismissing. Generalizationofourpresentre-
sultstomultiplefractionalvariationalintegralsisaninterestingandchallenging
openquestion. The recentresults provedin [3] may be useful to that objective.
The results of [20] are for fractional Riemann-Liouville cost integrals that
depend on a parameter α but not on fractional-order derivatives of order α as
we do here: the variational problems of [20] are defined for Lagrangians that
dependontheclassicalderivative,whileherewedealwithfractionalderivatives.
2
2 Preliminaries on Fractional Calculus
Inthis sectionwe fix notations by collecting the definitions of fractionalderiva-
tives in the sense of Riemann-Liouville, Caputo, and Riesz [2, 43, 45, 47].
Definition 1 (Riemann-Liouville fractional integrals). Let f be a continuous
functionintheinterval[a,b]. Fort∈[a,b],theleftRiemann-Liouvillefractional
integral Iαf(t) and the right Riemann-Liouville fractional integral Iαf(t) of
a t t b
order α, α>0, are defined by
1 t
Iαf(t)= (t−θ)α−1f(θ)dθ, (1)
a t Γ(α)
Za
1 b
Iαf(t)= (θ−t)α−1f(θ)dθ, (2)
t b Γ(α)
Zt
where Γ is the Euler gamma function.
Definition 2 (Riesz fractional integral). Let f be a continuous function in the
interval [a,b]. For t ∈ [a,b], the Riesz fractional integral RIαf(t) of order α,
a b
α>0, is defined by
1 b
RIαf(t)= |t−θ|α−1f(θ)dθ. (3)
a b 2Γ(α)
Za
Remark 3. From equations (1)–(3) it follows that
1
RIαf(t)= [ Iαf(t)+ Iαf(t)] . (4)
a b 2 a t t b
Definition 4 (fractional derivative in the sense of Riemann-Liouville). Let f
be a continuous function in the interval [a,b]. For t ∈ [a,b], the left Riemann-
Liouville fractional derivative Dαf(t) and the right Riemann-Liouville frac-
a t
tional derivative Dαf(t) of order α are defined by
t b
Dαf(t)=Dn In−αf(t)= 1 d n t(t−θ)n−α−1f(θ)dθ, (5)
a t a t Γ(n−α) dt
(cid:18) (cid:19) Za
Dαf(t)=(−D)n In−αf(t)= 1 −d n b(θ−t)n−α−1f(θ)dθ, (6)
t b t b Γ(n−α) dt
(cid:18) (cid:19) Zt
where n∈N is such that n−1≤α<n, and D is the usual derivative.
Definition 5 (fractional derivative in the sense of Caputo). Let f be a con-
tinuous function in [a,b]. For t ∈ [a,b], the left Caputo fractional derivative
CDαf(t) and the right Caputo fractional derivative CDαf(t) of order α are
a t t b
defined in the following way:
1 t d n
CDαf(t)= In−αDnf(t)= (t−θ)n−α−1 f(θ)dθ, (7)
a t a t Γ(n−α) dθ
Za (cid:18) (cid:19)
1 b d n
CDαf(t)= In−α(−D)nf(t)= (θ−t)n−α−1 − f(θ)dθ,
t b t b Γ(n−α) dθ
Zt (cid:18) (cid:19)
(8)
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where n∈N is such that n−1≤α<n.
Definition 6 (fractional derivatives in the sense of Riesz and Riesz-Caputo).
Letf beacontinuousfunctionin[a,b]. Fort∈[a,b],theRieszfractionalderiva-
tive RDαf(t) and the Riesz-Caputo fractional derivative RCDαf(t) of order α
a b a b
are defined by
RDαf(t)=DnRIn−αf(t)= 1 d n b|t−θ|n−α−1f(θ)dθ, (9)
a b a t Γ(n−α) dt
(cid:18) (cid:19) Za
1 b d n
RCDαf(t)=RIn−αDnf(t)= |t−θ|n−α−1 f(θ)dθ, (10)
a b a t Γ(n−α) dθ
Za (cid:18) (cid:19)
where n∈N is such that n−1≤α<n.
Remark 7. Using equations (4) and (5)–(10) it follows that
1
RDαf(t)= [ Dαf(t)+(−1)n Dαf(t)]
a b 2 a t t b
and
1
RCDαf(t)= CDαf(t)+(−1)nCDαf(t) .
a b 2 a t t b
In the particular case 0<α<1,(cid:2)we have: (cid:3)
1
RDαf(t)= [ Dαf(t)− Dαf(t)] (11)
a b 2 a t t b
and
1
RCDαf(t)= CDαf(t)− CDαf(t) . (12)
a b 2 a t t b
(cid:2) (cid:3)
Remark 8. If α=1, equalities (5)–(8) give the classical derivatives:
d d
D1f(t)= CD1f(t)= f(t), D1f(t)= CD1f(t)=− f(t).
a t a t dt t b t b dt
Substituting these quantities into (11) and (12), we obtain that
d
RD1f(t)= RCD1f(t)= f(t).
a b a b dt
3 Main Results
In2007aformulationoftheEuler-Lagrangeequationswasgivenforproblemsof
thecalculusofvariationswithfractionalderivativesinthesenseofRiesz-Caputo
[2]. Here we prove a fractional version of Noether’s theorem valid along the
Riesz-CaputofractionalEuler-Lagrangeextremals[2]. Forthatweintroducean
appropriatefractionaloperator that allow us to generalize the classicalconcept
of conservation law. Under the extended fractional notion of conservation law
we begin by proving in §3.1 a fractional Noether theorem without changing
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the time variable t, i.e., without transformation of the independent variable
(Theorem 21). In §3.2 we proceed with a time-reparameterization to obtain
the fractional Noether’s theorem in its general form (Theorem 24). Finally, in
§3.3 we consider more general fractional optimal control problems in the sense
of Riesz-Caputo, obtaining the corresponding fractional Noether’s theorem in
Hamiltonian form (Theorem 35).
3.1 On the Riesz-Caputo conservation of momentum
We begin by defining the fractional functional under consideration.
Problem 9 (The fractionalproblemofthe calculusofvariationsinthe senseof
Riesz-Caputo). The fractional problem of the calculus of variations in the sense
of Riesz-Caputo consists to find the stationary functions of the functional
b
I[q(·)]= L t,q(t),RCDαq(t) dt, (13)
a b
Za
(cid:0) (cid:1)
where [a,b] ⊂ R, a < b, 0 < α < 1, and the admissible functions q : t 7→ q(t)
andtheLagrangian L:(t,q,v )7→L(t,q,v )areassumedtobefunctionsofclass
l l
C2:
q(·)∈C2([a,b]; Rn);
L(·,·,·)∈C2([a,b]×Rn×Rn; R).
Along the work, we denote by ∂ L the partial derivative of L with respect
i
to its i-th argument, i=1,2,3.
Remark 10. When α = 1 the functional (13) is reduced to the classical func-
tional of the calculus of variations:
b
I[q(·)]= L(t,q(t),q˙(t))dt. (14)
Za
The next theorem summarizes the main result of [2].
Theorem11([2]). Ifq(·)isanextremizerof (13),thenitsatisfiesthefollowing
fractional Euler-Lagrangeequation in the sense of Riesz-Caputo:
∂ L t,q(t),RCDαq(t) −RDα∂ L t,q(t),RCDαq(t) =0 (15)
2 a b a b 3 a b
for all t∈[a,b].(cid:0) (cid:1) (cid:0) (cid:1)
Remark 12. The functional (13) involves Riesz-Caputo fractional derivatives
only. However, both Riesz-Caputoand Riesz fractional derivatives appear in the
fractional Euler-Lagrange equation (15).
Remark 13. Let α = 1. Then the fractional Euler-Lagrange equation in the
sense of Riesz-Caputo (15) is reduced to the classical Euler-Lagrange equation:
d
∂ L(t,q(t),q˙(t))− ∂ L(t,q(t),q˙(t))=0.
2 dt 3
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Theorem11leadstotheconceptoffractionalextremalinthesenseofRiesz-
Caputo.
Definition 14 (fractional extremal in the sense of Riesz-Caputo). A function
q(·) that is a solution of (15) is said to be a fractional Riesz-Caputo extremal
for functional (13).
In order to prove a fractional Noether’s theorem we adopt a technique used
in[23,30]. Forthat,webeginbyintroducingthenotionofvariationalinvariance
and by formulating a necessary condition of invariance without transformation
of the independent variable t.
Definition 15 (invariance of (13) without transformation of the independent
variable). Functional (13) is said to be invariant under an ε-parameter group
of infinitesimal transformations q¯(t)=q(t)+εξ(t,q(t))+o(ε) if
tb tb
L t,q(t),RCDαq(t) dt= L t,q¯(t),RCDαq¯(t) dt (16)
a b a b
Zta Zta
(cid:0) (cid:1) (cid:0) (cid:1)
for any subinterval [t ,t ]⊆[a,b].
a b
The next theorem establishes a necessary condition of invariance.
Theorem16(necessaryconditionofinvariance). Iffunctional (13)isinvariant
in the sense of Definition 15, then
∂ L t,q(t),RCDαq(t) ·ξ(t,q(t))+∂ L t,q(t),RCDαq(t) ·RCDαξ(t,q(t))=0.
2 a b 3 a b a b
(17)
(cid:0) (cid:1) (cid:0) (cid:1)
Remark 17. Let α = 1. From (17) we obtain the classical condition of in-
variance of the calculus of variations without transformation of the independent
variable t (cf., e.g., [41]):
∂ L(t,q,q˙)·ξ(t,q)+∂ L(t,q,q˙)·ξ˙(t,q)=0.
2 3
Proof. Having in mind that condition (16) is valid for any subinterval [t ,t ]⊆
a b
[a,b], we can get rid off the integralsigns in (16). Differentiating this condition
with respect to ε, then substituting ε = 0, and using the definitions and prop-
erties of the fractional derivatives given in Section 2, we arrive to the intended
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conclusion:
0=∂ L t,q(t),RCDαq(t) ·ξ(t,q)
2 a b
+∂ L(cid:0)t,q(t),RCDαq(t)(cid:1)· d 1 b|t−θ|n−α−1 d nq¯(θ)dθ
3 a b dε Γ(n−α) dθ
" Za (cid:18) (cid:19) #ε=0
(cid:0) (cid:1)
=∂ L t,q,RCDαq ·ξ(t,q)
2 a b
+(cid:0)∂ L t,q,RCD(cid:1) αq · d 1 b|t−θ|n−α−1 d nq(θ)dθ
3 a b dε Γ(n−α) dθ
" Za (cid:18) (cid:19)
(cid:0) (cid:1)
ε b d n
+ |t−θ|n−α−1 ξ(θ,q)dθ
Γ(n−α) dθ
Za (cid:18) (cid:19) #ε=0
=∂ L t,q,RCDαq ·ξ(t,q)
2 a b
+(cid:0)∂ L t,q,RCD(cid:1) αq · 1 b|t−θ|n−α−1 d nξ(θ,q)dθ
3 a b Γ(n−α) dθ
Za (cid:18) (cid:19)
=∂ L t,q,(cid:0)RCDαq ·ξ((cid:1)t,q)+∂ L t,q,RCDαq ·RCDαξ(t,q).
2 a b 3 a b a b
(cid:0) (cid:1) (cid:0) (cid:1)
Thefollowingdefinitionisusefulinordertointroduceanappropriateconcept
of fractional conserved quantity in the sense of Riesz-Caputo.
Definition 18. Given two functions f and g of class C1 in the interval [a,b],
we introduce the following operator:
Dγ(f,g)=g·RDγf +f ·RCDγg,
t a b a b
where t∈[a,b] and γ ∈R+.
0
Remark 19. Similar operators were used in [23, Definition 19] but involving
Riemann-Liouville fractional derivatives. We note that the new operator Dγ
t
proposed here involves both Riesz and Riesz-Caputo fractional derivatives.
Remark 20. In the classical context one has γ =1 and
d
D1(f,g)=f′·g+f ·g′ = (f ·g)=D1(g,f) .
t dt t
Roughlyspeaking,Dγ(f,g)isafractionalversionofthederivativeoftheproduct
t
off withg. Differentlyfrom theclassical context,inthefractional caseonehas,
in general, Dγ(f,g)6=Dγ(g,f).
t t
WenowprovethefractionalNoether’stheoreminthesenseofRiesz-Caputo
without transformation of the independent variable t.
Theorem 21 (Noether’s theorem in the sense of Riesz-Caputo without trans-
formationoftime). If functional (13) is invariant in the sense of Definition 15,
then
Dα ∂ L t,q(t),RCDαq(t) ,ξ(t,q(t)) =0 (18)
t 3 a b
along any fractional Riesz-Caputo extremal q(t), t∈[a,b] (Definition 14).
(cid:2) (cid:0) (cid:1) (cid:3)
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Remark 22. In the particular case when α = 1 we get from the fractional
conservation law in the sense of Riesz-Caputo (18) the classical Noether’s con-
servation law of momentum (cf., e.g., [30, 41]):
d
[∂ L(t,q(t),q˙(t))·ξ(t,q(t))]=0
dt 3
along any Euler-Lagrange extremal q(·) of (14). For this reason, we call the
fractional law (18) the fractional Riesz-Caputo conservation of momentum.
Proof. Using the fractional Euler-Lagrangeequation (15), we have:
∂ L t,q,RCDαq =RDα∂ L t,q,RCDαq . (19)
2 a b a b 3 a b
Replacing (19) in the(cid:0)necessary co(cid:1)ndition of inv(cid:0)ariance (17)(cid:1), we get:
RDα∂ L t,q,RCDαq ·ξ(t,q)+∂ L t,q,RCDαq ·RCDαξ(t,q)=0. (20)
a b 3 a b 3 a b a t
By definition(cid:0) of the oper(cid:1)ator Dγ(f,g) it(cid:0)results from(cid:1) (20) that
t
Dα ∂ L t,q,RCDαq ,ξ(t,q) =0.
t 3 a b
(cid:2) (cid:0) (cid:1) (cid:3)
3.2 The Noether theorem in the sense of Riesz-Caputo
The next definition gives a more general notion of invariance for the integral
functional (13). The main result of this section, the Theorem 24, is formulated
with the help of this definition.
Definition 23 (invariance of (13)). The integral functional (13) is said to be
invariant under the one-parameter group of infinitesimal transformations
t¯=t+ετ(t,q(t))+o(ε),
(q¯(t)=q(t)+εξ(t,q(t))+o(ε),
if
tb t¯(tb)
L t,q(t),RCDαq(t) dt= L t¯,q¯(t¯),RCDαq¯(t¯) dt¯
a b a b
Zta Zt¯(ta)
(cid:0) (cid:1) (cid:0) (cid:1)
for any subinterval [t ,t ]⊆[a,b].
a b
Our next theorem gives a generalizationof Noether’s theorem for fractional
problems of the calculus of variations in the sense of Riesz-Caputo.
Theorem 24 (Noether’s fractional theorem in the sense of Riesz-Caputo). If
the integral functional (13) is invariant in the sense of Definition 23, then
Dα ∂ L t,q,RCDαq ,ξ(t,q)
t 3 a t
(cid:2) +D(cid:0)tα L t,q,aRC(cid:1)Dtαq −(cid:3)α∂3L t,q,aRCDtαq ·RaCDbαq,τ(t,q) =0 (21)
along any frac(cid:2)tio(cid:0)nal Riesz-Ca(cid:1)puto extr(cid:0)emal q(·). (cid:1) (cid:3)
8
Remark 25. In the particular case α = 1 we obtain from (21) the classical
Noether’s conservation law (cf., e.g., [30, 41]):
d
[∂ L(t,q,q˙)·ξ(t,q)+(L(t,q,q˙)−∂ L(t,q,q˙)·q˙)τ(t,q)]=0
dt 3 3
along any Euler-Lagrange extremal q(·) of (14).
Proof. Our proof is an extension of the method used in [30] to prove the classi-
cal Noether’s theorem. For that we reparameterize the time (the independent
variable t) with a Lipschitzian transformation
[a,b]∋t7−→σf(λ)∈[σ ,σ ]
a b
that satisfies
′ dt(σ)
t = =f(λ)=1 if λ=0. (22)
σ dσ
In this way one reduces (13) to an autonomous integral functional
I¯[t(·),q(t(·))]= σbL t(σ),q(t(σ)),RCDα q(t(σ)) t′ dσ, (23)
σa σb σ
Zσa
(cid:0) (cid:1)
wheret(σ )=aandt(σ )=b. Usingthedefinitionsandpropertiesoffractional
a b
derivatives given in Section 2, we get successively that
RCDα q(t(σ))= 1 f(bλ) |σf(λ)−θ|n−α−1 d nq θf−1(λ) dθ
σa σb Γ(n−α) a dθ(σ)
Zf(λ) (cid:18) (cid:19)
(cid:0) (cid:1)
= (t′σ)−α (t′σb)2 |σ−s|n−α−1 d nq(s)ds
Γ(n−α) ds
Z(t′σa)2 (cid:18) (cid:19)
=(t′ )−α RCDαq(σ) χ= a , ω = b .
σ χ ω (t′ )2 (t′ )2
(cid:18) σ σ (cid:19)
We then have
I¯[t(·),q(t(·))]= σbL t(σ),q(t(σ)),(t′ )−α RCDαq(σ)q(σ) t′ dσ
σ χ ω σ
Zσa (cid:16) (cid:17)
=. σbL¯ t(σ),q(t(σ)),t′ , RCDαq(σ) dσ
f σ χ ω
Zσa (cid:16) (cid:17)
b
= L t,q(t),RCDαq(t) dt
a b
Za
=I[q(·)](cid:0). (cid:1)
Ifthe integralfunctional(13)is invariantin the sense ofDefinition 23, then the
integralfunctional(23)isinvariantinthe senseofDefinition15. Itfollowsfrom
Theorem 21 that
∂
Dα ∂ L¯ ,ξ +Dα L¯ ,τ =0 (24)
t 4 f t ∂t′ f
(cid:20) σ (cid:21)
(cid:2) (cid:3)
9
is a fractional conserved law in the sense of Riesz-Caputo. For λ = 0 the
condition (22) allows us to write that
RCDαq(σ)=RC Dαq(t),
χ ω a b
and therefore we get that
∂ L¯ =∂ L (25)
4 f 3
and
∂ L¯ =∂ L¯ · ∂ (t′σ)−α ω|σ−s|n−α−1 d nq(s)ds t′ +L
∂t′ f 4 f ∂t′ Γ(n−α) ds σ
σ σ " Zχ (cid:18) (cid:19) #
=∂ L¯ · −α(t′σ)−α−1 ω|σ−s|n−α−1 d nq(s)ds t′ +L (26)
4 f Γ(n−α) ds σ
" Zχ (cid:18) (cid:19) #
=−α∂ L·RCDαq+L.
3 a b
Substituting the quantities (25) and (26) into (24), we obtain the fractional
conservation law in the sense of Riesz-Caputo (21).
3.3 Optimal control of Riesz-Caputo fractional systems
We now adopt the Hamiltonian formalism in order to generalize the Noether
type results found in [14, 49] for the more generalcontext of fractional optimal
control in the sense of Riesz-Caputo. For this, we make use of our Noether’s
Theorem 24 and the standard Lagrange multiplier technique (cf. [14]). The
fractional optimal control problem in the sense of Riesz-Caputo is introduced,
without loss of generality, in Lagrange form:
b
I[q(·),u(·)]= L(t,q(t),u(t))dt−→min, (27)
Za
subject to the fractional differential system
RCDαq(t)=ϕ(t,q(t),u(t)) (28)
a a
and initial condition
q(a)=q . (29)
a
The Lagrangian L : [a,b]×Rn ×Rm → R and the fractional velocity vector
ϕ:[a,b]×Rn×Rm →RnareassumedtobefunctionsofclassC1 withrespectto
alltheir arguments. We alsoassume,withoutlossofgenerality,that0<α≤1.
Inconformitywiththecalculusofvariations,weareconsideringthatthecontrol
functions u(·) take values on an open set of Rm.
Definition 26. The fractional differential system (28) is said to be a fractional
control system in the sense of Riesz-Caputo.
10