Discrete-time fractional variational problems

Signal Processing 91 (2011) 513–524

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Discrete-time fractional variational problems Nuno R.O. Bastos a, Rui A.C. Ferreira b, Delfim F.M. Torres c, a

Department of Mathematics, ESTGV, Polytechnic Institute of Viseu, 3504-510 Viseu, Portugal Faculty of Engineering and Natural Sciences, Lusophone University of Humanities and Technologies, 1749-024 Lisbon, Portugal c Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal b

a r t i c l e in fo

abstract

Available online 8 May 2010

We introduce a discrete-time fractional calculus of variations on the time scale ðhZÞa , a 2 R,h 4 0. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler–Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of the discrete-derivatives are integer values, and that they converge to the fractional continuous-time solutions when h tends to zero. Our Legendre type condition is useful to eliminate false candidates identified via the Euler–Lagrange fractional equation. & 2010 Elsevier B.V. All rights reserved.

Keywords: Fractional difference calculus Calculus of variations Fractional summation by parts Euler–Lagrange equation Natural boundary conditions Legendre necessary condition Time scale hZ

1. Introduction The fractional calculus (calculus with derivatives of arbitrary order) is an important research field in several different areas such as physics (including classical and quantum mechanics as well as thermodynamics), chemistry, biology, economics, and control theory [3,10,40,42, 48]. It has its origin more than 300 years ago when L’Hopital asked Leibniz what should be the meaning of a derivative of non-integer order. After that episode several more famous mathematicians contributed to the development of fractional calculus: Abel, Fourier, Liouville, Riemann, Riesz, just to mention a few names [30,47]. In the last decades, considerable research has been done in fractional calculus. This is particularly true in the area of the calculus of variations, which is being subject to intense investigations during the last few years [10,11,44,45]. Applications include fractional variational principles in mechanics and physics, quantization, control theory, and description of conservative, non-conservative, and constrained systems [10,15,16,45]. Roughly speaking, the

 Corresponding author.

E-mail addresses: [email protected] (N.R.O. Bastos), [email protected] (R.A.C. Ferreira), delfi[email protected] (D.F.M. Torres). 0165-1684/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2010.05.001

classical calculus of variations and optimal control is extended by substituting the usual derivatives of integer order by different kinds of fractional (non-integer) derivatives. It is important to note that the passage from the integer/classical differential calculus to the fractional one is not unique because we have at our disposal different notions of fractional derivatives. This is, as argued in [10,44], an interesting and advantage feature of the area. Most part of investigations in the fractional variational calculus are based on the replacement of the classical derivatives by fractional derivatives in the sense of Riemann–Liouville, Caputo, Riesz, and Jumarie [1,4,10, 27]. Independently of the chosen fractional derivatives, one obtains, when the fractional order of differentiation tends to an integer order, the usual problems and results of the calculus of variations. Although the fractional Euler– Lagrange equations are obtained in a similar manner as in the standard variational calculus [44], some classical results are extremely difficult to be proved in a fractional context. This explains, for example, why a fractional Legendre type condition is absent from the literature of fractional variational calculus. In this work we give a first result in this direction (cf. Theorem 3.6). Despite its importance in applications, less is known for discrete-time fractional systems [44]. In [39] Miller and Ross define a fractional sum of order n 4 0 via the

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solution of a linear difference equation. They introduce it as (see Section 2 for the notations used here)

Dn f ðtÞ ¼

tn 1 X ðtsðsÞÞðn1Þ f ðsÞ: GðnÞ s ¼ a

difficult and challenging problem since our proofs deeply rely on the fact that in T ¼ ðhZÞa the graininess function is a constant.

ð1Þ

Definition (1) is analogous to the Riemann–Liouville fractional integral Z x 1 n ðxsÞn1 f ðsÞ ds a Dx f ðxÞ ¼ GðnÞ a of order n 4 0, which can be obtained via the solution of a linear differential equation [39,40]. Basic properties of the operator Dn in (1) were obtained in [39]. More recently, Atici and Eloe introduced the fractional difference of order a 4 0 by Da f ðtÞ ¼ Dm ðDam f ðtÞÞ, where m is the integer part of a, and developed some of its properties that allow to obtain solutions of certain fractional difference equations [8,9]. The fractional differential calculus has been widely developed in the past few decades due mainly to its demonstrated applications in various fields of science and engineering [30,40,43]. The study of necessary optimality conditions for fractional problems of the calculus of variations and optimal control is a fairly recent issue attracting an increasing attention—see [1,2,7,22,23, 25,26,41] and references therein—but available results address only the continuous-time case. It is well known that discrete analogues of differential equations can be very useful in applications [13,29,31] and that fractional Euler–Lagrange differential equations are extremely difficult to solve, being necessary to discretize them [2,11]. Therefore, it is pertinent to develop a fractional discretetime theory of the calculus of variations for the time scale ðhZÞa ,h4 0 (cf. definitions in Section 2). Computer simulations show that this time scale is particularly interesting because when h tends to zero one recovers previous fractional continuous-time results. Our objective is twofold. On one hand we proceed to develop the theory of fractional difference calculus, namely, we introduce the concept of left and right fractional sum/ difference (cf. Definition 2.8). On the other hand, we believe that the present work will potentiate research not only in the fractional calculus of variations but also in solving fractional difference equations, specifically, fractional equations in which left and right fractional differences appear. Because the theory of fractional difference calculus is still in its infancy [8,9,39], the paper is self-contained. In Section 2 we introduce notations, we give necessary definitions, and prove some preliminary results needed in the sequel. Main results of the paper appear in Section 3: we prove a fractional formula of h-summation by parts (Theorem 3.2), and necessary optimality conditions of first and second order (Theorems 3.5 and 3.6, respectively) for the proposed h-fractional problem of the calculus of variations (17). Section 4 gives some illustrative examples, and we end the paper with Section 5 of conclusions and future perspectives. The results of the paper are formulated using standard notations of the theory of time scales [20,32,33]. It remains an interesting open question how to generalize the present results to an arbitrary time scale T. This is a

2. Preliminaries We begin by recalling the main definitions and properties of time scales (cf. [18,20] and references therein). A non-empty closed subset of R is called a time scale and is denoted by T. The forward jump operator s : T-T is defined by sðtÞ ¼ inffs 2 T : s 4tg for all t 2 T, while the backward jump operator r : T-T is defined by rðtÞ ¼ supfs 2 T : so tg for all t 2 T, with inf| ¼ supT (i.e., sðMÞ ¼ M if T has a maximum M) and sup| ¼ inf T (i.e., rðmÞ ¼ m if T has a minimum m). A point t 2 T is called right-dense, right-scattered, left-dense, or left-scattered, if sðtÞ ¼ t, sðtÞ 4t, rðtÞ ¼ t, or rðtÞ ot, respectively. Through~ with a o b and T ~ a time out the text we let T ¼ ½a,b \ T 2 scale. We define Tk ¼ T\ðrðbÞ,b, Tk ¼ ðTk Þk and more n n1 generally Tk ¼ ðTk Þk , for n 2 N. The following standard notation is used for s (and rÞ: s0 ðtÞ ¼ t, sn ðtÞ ¼ ðs3sn1 ÞðtÞ,n 2 N. The graininess function m : T-½0,1Þ is defined by mðtÞ ¼ sðtÞt for all t 2 T. A function f : T-R is said to be delta differentiable at t 2 Tk if there is a number f D ðtÞ such that for all e 4 0 there exists a neighborhood U of t (i.e., U ¼ ðtd,t þ dÞ \ T for some d 40Þ such that jf ðsðtÞÞf ðsÞf D ðtÞðsðtÞsÞj r ejsðtÞsj

for all s 2 U:

f D ðtÞ

We call the delta derivative of f at t. The rth-delta derivative ðr 2 NÞ of f is defined to be the function r r r r1 f D : Tk -R, provided f D is delta differentiable on Tk . For delta differentiable f and g and for an arbitrary time scale T the next formulas hold: f s ðtÞ ¼ f ðtÞ þ mðtÞf D ðtÞ and ðfgÞD ðtÞ ¼ f D ðtÞg s ðtÞ þ f ðtÞg D ðtÞ ¼ f D ðtÞgðtÞ þ f s ðtÞg D ðtÞ,

ð2Þ

where we abbreviate f 3s by f s . 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. The set of all rd-continuous functions is denoted by Crd and the set of all delta differentiable functions with rdcontinuous derivative by C1rd. It is known that rdcontinuous functions possess an antiderivative, i.e., there 1 exists a function F 2 Crd with F D ¼ f . The delta integral is Rb then defined by a f ðtÞDt ¼ FðbÞFðaÞ. It satisfies the R sðtÞ equality t f ðtÞDt ¼ mðtÞf ðtÞ. We make use of the following properties of the delta integral: Lemma 2.1 (cf. Bohner and Peterson [20, Theorem 1.77]). If a,b 2 T and f ,g 2 Crd , then 1. R b f ðsðtÞÞg D ðtÞDt ¼ ðfgÞðtÞjt ¼ b  R b f D ðtÞgðtÞDt; t¼a a a 2. R b f ðtÞg D ðtÞDt ¼ ðfgÞðtÞjt ¼ b  R b f D ðtÞgðsðtÞÞDt. a

t¼a

a

One way to approach the Riemann–Liouville fractional calculus is through the theory of linear differential equations [43]. Miller and Ross [39] use an analogous methodology to introduce fractional discrete operators for the case T ¼ Za ¼ fa,a þ1,a þ 2, . . .g,a 2 R. Here we go a step further: we use the theory of time scales in order to

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

introduce fractional discrete operators to the more general case T ¼ ðhZÞa ¼ fa,a þ h,a þ2h, . . .g,a 2 R,h 4 0. For n 2 N0 and rd-continuous functions pi : TR,1 ri r n, let us consider the nth order linear dynamic equation n X

515

our results to an arbitrary time scale T, with the graininess function m depending on time, is not clear and remains a challenging question. Let a 2 R and h4 0, ðhZÞa ¼ fa,a þ h,a þ 2h, . . .g, and b= a+ kh for some k 2 N. We have sðtÞ ¼ t þ h, rðtÞ ¼ We put T ¼ ½a,b \ ðhZÞa , so that th, mðtÞ  h. 2

ð3Þ

Tk ¼ ½a, rðbÞ \ ðhZÞa and Tk ¼ ½a, r2 ðbÞ \ ðhZÞa . The delta

A function y : T-R is said to be a solution of Eq. (3) on T n provided y is n times delta differentiable on Tk and kn satisfies Ly(t)= 0 for all t 2 T .

derivative coincides in this case with the forward h-difference: f D ðtÞ ¼ ðf s ðtÞ f ðtÞÞ=mðtÞ. If h= 1, then we have the usual discrete forward difference Df ðtÞ. The delta integral gives the h-sum (or h-integral) of f: Rb Pb=h1 f ðkhÞh. If we have a function f of two a f ðtÞDt ¼ k ¼ a=h

Ly ¼ 0

n

where Ly ¼ yD þ

ni

pi yD :

i¼1

Lemma 2.2 (Bohner and Peterson [20, p. z ¼ ðz1 , . . . ,zn Þ : T-Rn satisfies for all t 2 Tk 0 0 1 0 ... B ^ 0 1 & B B & & zD ¼ AðtÞzðtÞ where A ¼ B B ^ B 0 ... ... 0 @ pn

...

...

p2

239]). If 0

1

C C C C C 1 C A p1 ^ 0

ð4Þ then y=z1 is a solution of Eq. (3). Conversely, if y solves (3) n1 on T, then z ¼ ðy,yD , . . . ,yD Þ : T-R satisfies (4) for all kn t2T . Definition 2.3 (Bohner and Peterson [20, p. 239]). We say that Eq. (3) is regressive provided I þ mðtÞAðtÞ is invertible for all t 2 Tk , where A is the matrix in (4). Definition 2.4 (Bohner and Peterson [20, p. 250]). We n define the Cauchy function y : T  Tk -R for the linear n dynamic equation (3) to be, for each fixed s 2 Tk , the solution of the initial value problem Ly ¼ 0,

Di

Dn1

y ðsðsÞ,sÞ ¼ 0, 0 r ir n2, y

ððsðsÞ,sÞ ¼ 1:

ð5Þ

Theorem 2.5 (Bohner and Peterson [20, p. 251]). Suppose {y1,y,yn} is a fundamental system of the regressive equation (3). Let f 2 Crd . Then the solution of the initial value problem

Definition 2.6. For arbitrary x,y 2 R the h-factorial function is defined by x  G þ1 ðyÞ y h , xh :¼ h  x G þ 1y h where G is the well-known Euler gamma function, and we use the convention that division at a pole yields zero. Remark 2.1. For h =1, and in accordance with the previous literature (1), we write x(y) to denote x(y) h . Proposition 2.1. For the time-scale T ¼ ðhZÞa one has

Di

y ðt0 Þ ¼ 0, 0 r ir n1 Rt is given by yðtÞ ¼ t0 yðt,sÞf ðsÞDs, where y(t,s) is the Cauchy function for (3). Ly ¼ f ðtÞ,

variables, f(t,s), its partial forward h-differences will be denoted by Dt,h and Ds,h , respectively. We will make P use of the standard conventions tc1 ¼ c f ðtÞ ¼ 0,c 2 Z, and Q1 f ðiÞ ¼ 1. Often, left fractional delta integration (resp., i¼0 right fractional delta integration) of order n 4 0 is denoted n n by a D t f ðtÞ (resp. t Db f ðtÞÞ. Here, similarly as in Ross et al. [46], where the authors omit the subscript t on the operator (the operator itself cannot depend on t), we write n n a Dh f ðtÞ (resp. h Db f ðtÞÞ. Before giving an explicit formula for the generalized polynomials Hk on hZ we introduce the following definition:

It is known that yðt,sÞ :¼ Hn1 ðt, sðsÞÞ is the Cauchy n function for yD ¼ 0, where Hn  1 is a time scale generalized polynomial [20, Example 5.115]. The generalized polynomials Hk are the functions Hk : T2 -R, k 2 N0 , defined recursively as follows: Z t H0 ðt,sÞ  1, Hk þ 1 ðt,sÞ ¼ Hk ðt,sÞDt, k ¼ 1,2, . . .

Hk ðt,sÞ :¼

ðtsÞðkÞ h k!

for all s,t 2 T and k 2 N0 :

ð6Þ

To prove (6) we use the following technical lemma. Throughout the text the basic property Gðx þ1Þ ¼ xGðxÞ of the gamma function will be frequently used. Lemma 2.7. Let s 2 T. Then, for all t 2 Tk one has ( ) þ 1Þ ðtsÞðk ðtsÞðkÞ h h : Dt,h ¼ ðkþ 1Þ! k!

s

for all s,t 2 T. If we let HkD ðt,sÞ denote, for each fixed s, the derivative of Hk(t,s) with respect to t, then (cf. [20, p. 38]) HkD ðt,sÞ ¼ Hk1 ðt,sÞ

for k 2 N, t 2 Tk :

From now on we restrict ourselves to the time scale

T ¼ ðhZÞa ,h 4 0, for which the graininess function is the constant h. Our main goal is to propose and develop a discrete-time fractional variational theory in T ¼ ðhZÞa . We borrow the notations from the recent calculus of variations on time scales [18,24,32]. How to generalize

Proof. The equality follows by direct computations: ( ) ( ) ðk þ 1Þ ðtsÞhðk þ 1Þ ðtsÞhðk þ 1Þ 1 ðsðtÞsÞh ¼  h ðk þ 1Þ! ðk þ 1Þ! ðkþ 1Þ!   hk þ 1 Gððt þ hsÞ=h þ 1Þ GððtsÞ=h þ 1Þ  ¼ hðk þ 1Þ! Gððt þ hsÞ=h þ 1ðk þ 1ÞÞ GððtsÞ=h þ 1ðk þ 1ÞÞ   hk ððtsÞ=h þ 1ÞGððtsÞ=h þ 1Þ GððtsÞ=h þ 1Þ  ¼ ðk þ 1Þ! ððtsÞ=hkÞGððtsÞ=hkÞ GððtsÞ=hkÞ   ðtsÞðkÞ hk GððtsÞ=h þ 1Þ h : & ¼ ¼ k! GððtsÞ=h þ 1kÞ k!

Dt,h

516

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

Proof of Proposition 2.1. We proceed by mathematical induction. For k= 0     ts ts þ1 þ1 G G 1 h h ¼   ¼ 1: H0 ðt,sÞ ¼ h0  ts ts 0! þ 10 þ1 G G h h Assume that (6) holds for k replaced by m. Then by Lemma 2.7 Hm þ 1 ðt,sÞ ¼

Z

t

Hm ðt,sÞDt ¼ s

Z s

t

þ 1Þ ðtsÞðmÞ ðtsÞðm h h , Dt ¼ m! ðm þ1Þ!

which is (6) with k replaced by m+ 1.

&

following meaning: t=h Xn

GðkÞ ¼ Gða=hÞ þ Gða=h þ 1Þ þ Gða=hþ 2Þ þ    þGðt=hnÞ,

k ¼ ha

þ kh þ nhg with where t 2 fa þ nh,a þ h þ nh,a þ 2h þ nh, . . . ,a |fflffl{zfflffl} k 2 N. b Lemma 2.9. Let n 40 be an arbitrary positive real number. n For any t 2 T we have: (i) limn-0 a D h f ðt þ nhÞ ¼ f ðtÞ; (ii) n limn-0 h D f ðt n hÞ ¼ f ðtÞ. b Proof. Since a

n D h f ðt þ nhÞ ¼

Let y1(t),y,yn(t) be n linearly independent solutions of n the linear homogeneous dynamic equation yD ¼ 0. From Theorem 2.5 we know that the solution of (5) (with L ¼ Dn and t0 =a) is yðtÞ ¼ Dn f ðtÞ ¼

Z

t

a

¼

ðtsðsÞÞhðn1Þ f ðsÞDs GðnÞ

ð7Þ

Definition 2.8. Let a 2 R,h4 0, b= a +kh with k 2 N, and put T ¼ ½a,b \ ðhZÞa . Consider f 2 F T . The left and right fractional h-sum of order n 4 0 are, respectively, the ~7¼ operators Dn : F T -F þ and Dn : F T -F ~  , T b

Tn

n

ft 7 nh : t 2 Tg, defined by Z sðtnhÞ 1 n ðtsðsÞÞðhn1Þ f ðsÞDs a Dh f ðtÞ ¼ GðnÞ a ¼

1

GðnÞ k ¼ a=h

¼ hn f ðtÞ þ

rX ðtÞ=h n ðt þ nhsðkhÞÞðhn1Þ f ðkhÞh, Gðn þ 1Þ k ¼ a=h

h

n n D b f ðtnhÞ ¼ h f ðtÞ þ

n Gðn þ1Þ

Z sðbÞ sðtÞ

ðs þ nhsðtÞÞðhn1Þ f ðsÞDs: ð8Þ

Note that function t-ðDn f ÞðtÞ is defined for t =a + nh mod(h) while function t-f ðtÞ is defined for t = a mod(h). Extending (7) to any positive real value n, and having as an analogy the continuous left and right fractional derivatives [40], we define the left fractional h-sum and the right fractional h-sum as follows. We denote by F T the set of all real valued functions defined on a given time scale T.

t=h Xn

t=h 1 X ðt þ nhsðkhÞÞðhn1Þ f ðkhÞh GðnÞ k ¼ a=h

any t 2 T and for any n Z0 we define :¼ h D0b f ðtÞ :¼ f ðtÞ and write Z t n n n ðt þ nhsðsÞÞðhn1Þ f ðsÞDs, a Dh f ðt þ nhÞ ¼ h f ðtÞ þ Gðn þ 1Þ a

t=hn 1 X ðtsðkhÞÞðn1Þ f ðkhÞh h GðnÞ k ¼ a=h Z sðtnhÞ 1 ¼ ðtsðsÞÞhðn1Þ f ðsÞDs: GðnÞ a

h

ðt þ nhsðsÞÞðhn1Þ f ðsÞDs

For

Dn f ðtÞ ¼

T~ n

a

0 a Dh f ðtÞ

Since yDi ðaÞ ¼ 0, i= 0,y,n  1, then we can write that

h

Z sðtÞ

n it follows that limn-0 a D h f ðt þ nhÞ ¼ f ðtÞ. The proof of (ii) is similar. &

t=h1 1 X ðtsðkhÞÞhðn1Þ f ðkhÞh: GðnÞ k ¼ a=h

a

¼

1 GðnÞ

ðtsðkhÞÞðhn1Þ f ðkhÞh,

Z sðbÞ 1 n ðssðtÞÞðhn1Þ f ðsÞDs h Db f ðtÞ ¼ GðnÞ t þ nh b=h X 1 ðkhsðtÞÞhðn1Þ f ðkhÞh: ¼ GðnÞ k ¼ t=h þ n Remark 2.2. In Definition 2.8 we are using summations with limits that are reals. For example, the summation n that appears in the definition of operator a D has the h

Theorem 2.10. Let f 2 F T and n Z0. For all t 2 Tk we have n D a Dh f ðt þ

nhÞ ¼ ða Dh n f ðt þ nhÞÞD n ðt þ nhaÞðhn1Þ f ðaÞ:  Gðn þ 1Þ

ð9Þ

To prove Theorem 2.10 we make use of a technical lemma: Lemma 2.11. Let t 2 Tk . The following equality holds for all s 2 Tk :

Ds,h ððt þ nhsÞðhn1Þ f ðsÞÞÞ ¼ ðt þ nhsðsÞÞðhn1Þ f D ðsÞðv1Þðt þ nhsðsÞÞðhn2Þ f ðsÞ: ð10Þ Proof. Direct calculations give the intended result:

Ds,h ððt þ nhsÞðhn1Þ f ðsÞÞ ¼ Ds,h ððt þ nhsÞhðn1Þ Þf ðsÞ þ ðt þ nhsðsÞÞhðn1Þ f D ðsÞ   2 t þ nhsðsÞ þ1 G f ðsÞ 6 h 6hn1   ¼ t þ nhsðsÞ h 4 þ 1ðn1Þ G h   3 t þ nhs þ1 G 7 h ðn1Þ D 7 f ðsÞ hn1  5 þ ðt þ nhsðsÞÞh t þ nhs G þ 1ðn1Þ h

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

2

  33 2  t þ nhs t þ nhs þ1 G G 77 h h 7      7 55 ts ts þ1 þ2 G G h h

517

Theorem 2.12. Let f 2 F T and n Z0. For all t 2 Tk we have

6 n2 6 6 ¼ f ðsÞ6 4h 4

n n D h DrðbÞ f ðt nhÞ ¼ Gðn þ 1Þðbþ nh

sðtÞÞðhn1Þ f ðbÞþ ðh Db n f ðtnhÞÞD : ð13Þ

ðn1Þ

þ ðt þ nhsðsÞÞh f D ðsÞ   t þ nhsh þ1 G h  ððn1ÞÞ ¼ f ðsÞhn2  ts þ nhh G þ 1ðn2Þ h

Proof. From (11) we obtain from integration by parts (item 2 of Lemma 2.1) that n D h DrðbÞ f ðt

nhÞ ¼

þ ðt þ nhsðsÞÞðhn1Þ f D ðsÞ ¼ ðn1Þðt þ nhsðsÞÞðhn2Þ f ðsÞ þ ðt þ nhsðsÞÞðhn1Þ f D ðsÞ, where the first equality follows directly from (2).

&

Remark 2.3. Given an arbitrary t 2 Tk it is easy to prove, in a similar way as in the proof of Lemma 2.11, the following equality analogous to (10): for all s 2 Tk ðtÞÞðhn1Þ f ðsÞÞÞ

Ds,h ððs þ nhs

n D and this shows the We now compute ðh D b f ðtnhÞÞ veracity of (13): n D ðh D b f ðtnhÞÞ

¼ hn f D ðtÞ þ

ðtÞÞðhn2Þ f s ðsÞ þðs þ

¼ ðn1Þðs þ nhs

ðtÞÞðhn1Þ f D ðsÞ:

nhs

Z sðbÞ

ð11Þ



n hGðn þ 1Þ

n D a Dh f ðt þ

nhÞ ¼ hn f D ðtÞ þ Z

t a

ðt þ nhsðsÞÞhðn1Þ f D ðsÞDs

¼ hn f D ðtÞ þ þ

¼ þ

n Gðn þ 1Þ

n Gðn þ 1Þ

n Gðn þ1Þ

Z sðtÞ a

t ½ðt þ nhsÞðhn1Þ f ðsÞss ¼ ¼a

ðn1Þðt þ nhsðsÞÞðhn2Þ f ðsÞDs

nðt þ nhaÞðhn1Þ f ðaÞ þhn f D ðtÞ þ nhn1 f ðtÞ Gðn þ 1Þ n Gðn þ1Þ

We now compute veracity of (9):

Z a

t

ðn1Þðt þ nhsðsÞÞðhn2Þ f ðsÞDs:

n ða D h f ðt þ

D

nhÞÞ

and this shows the

n D ða D h f ðt þ nhÞÞ Z sðtÞ 1 n n h f ðsðtÞÞ þ ¼ ðsðtÞ þ nhsðsÞÞðhn1Þ f ðsÞDs h Gðn þ1Þ a

Z t n ðt þ nhsðsÞÞðhn1Þ f ðsÞDs hn f ðtÞ Gðn þ 1Þ a Z t n n D ðsðtÞ þ nhsðsÞÞðhn1Þ f ðsÞDs ¼ h f ðtÞ þ hGðn þ 1Þ a

Z t ðt þ nhsðsÞÞðhn1Þ f ðsÞDs þhn1 nf ðtÞ ¼ hn f D ðtÞ  a Z t n D ððt þ nhsðsÞÞhðn1Þ Þf ðsÞDs þ hn1 nf ðtÞ þ Gðn þ1Þ a t,h Z t n ðn1Þðt þ nhsðsÞÞðhn2Þ f ðsÞDs ¼ hn f D ðtÞ þ Gðn þ 1Þ a þ nhn1 f ðtÞ: &

Follows the counterpart of Theorem 2.10 for the right fractional h-sum:

"Z

sðbÞ

s2 ðtÞ

ðs þ nhs2 ðtÞÞÞðhn1Þ f ðsÞDs #

ðtÞÞðhn1Þ f ðsÞDs

ðs þ nhs

s2 ðtÞ

Proof of Theorem 2.10. From Lemma 2.11 we obtain that

nðb þ nhsðtÞÞðhn1Þ f ðbÞ þ hn f D ðtÞ Gðn þ 1Þ Z b n ðn1Þ nhn1 f ðsðtÞÞ Gðn þ 1Þ sðtÞ ðs þ nhsðtÞÞðhn2Þ f s ðsÞDs:

nhn1 f ðsðtÞÞ

Z

sðbÞ n D ððs þ nhsðtÞÞðhn1Þ Þf ðsÞDs Gðn þ 1Þ s2 ðtÞ t,h nhn1 f ðsðtÞÞ Z sðbÞ n ¼ hn f D ðtÞ ðn1Þðsþ nhs2 ðtÞÞðhn2Þ f ðsÞDs Gðn þ1Þ s2 ðtÞ nhn1 f ðsðtÞÞ Z b n ðn1Þðs þ nhsðtÞÞðhn2Þ f s ðsÞDs ¼ hn f D ðtÞ Gðn þ1Þ sðtÞ & nhn1 f ðsðtÞÞ:

¼ hn f D ðtÞ þ

Definition 2.13. Let 0 o a r 1 and set g :¼ 1a. The left fractional difference a Dah f ðtÞ and the right fractional difference h Dab f ðtÞ of order a of a function f 2 F T are defined as a

a Dh f ðtÞ

g

:¼ ða Dh f ðt þ ghÞÞD

and

a

h Db f ðtÞ

g

:¼ ðh Db f ðtghÞÞD

for all t 2 Tk . 3. Main results Our aim is to introduce the h-fractional problem of the calculus of variations and to prove corresponding necessary optimality conditions. In order to obtain an Euler– Lagrange type equation (cf. Theorem 3.5) we first prove a fractional formula of h-summation by parts.

3.1. Fractional h-summation by parts A big challenge was to discover a fractional h-summation by parts formula within the time scale setting. Indeed, there is no clue of what such a formula should be. We found it eventually, making use of the following lemma. Lemma 3.1. Let f and k be two functions defined on Tk and 2 2 Tk , respectively, and g a function defined on Tk  Tk . The

518

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

following equality holds: "Z Z t

Z b Z rðbÞ f ðtÞ gðt,sÞkðsÞDs Dt ¼ kðtÞ a

a

Z #

b

a

Z

gðs,tÞf ðsÞDs Dt:

g

f ðtÞa Dh g D ðt þ ghÞDt b

þ

sðtÞ

a

b

¼

a

Proof. Consider the matrices R ¼ ½f ða þ hÞ,f ða þ2hÞ, . . . , f ðbhÞ, 2 6 6 C1 ¼ 6 6 4

3

gða þh,aÞkðaÞ gðaþ 2h,aÞkðaÞ þgða þ2h,a þ hÞkða þhÞ ^ gðbh,aÞkðaÞ þ gðbh,a þ hÞkða þhÞ þ    þgðbh,b2hÞkðb2hÞ

7 7 7, 7 5

2

3 2 3 0 gða þ h,aÞ 6 7 6 7 6 gða þ 2h,aÞ 7 6 gða þ 2h,a þ hÞ 7 7, C3 ¼ 6 7, C2 ¼ 6 6 7 6 7 ^ ^ 4 5 4 5 gðbh,a þhÞ gðbh,aÞ 2 3 0 6 7 0 6 7 7: C4 ¼ 6 6 7 ^ 4 5 gðbh,b2hÞ

b

f ðtÞ

Z

a

t

b=h1 X

#

b=h2 X

gðjh,b2hÞf ðjhÞ

¼

kðtÞ

"Z

gðs,tÞf ðsÞDs Dt:

a

¼ hg f ðrðbÞÞgðbÞhg f ðaÞgðaÞ þ Z

g gðaÞ Gðg þ 1Þ Z

a

b

 sðaÞ

&

Z rðbÞ a

b

ðg1Þ

a

h DrðbÞ f ðtÞg

ðg1Þ

ðt þ ghaÞh

ðt þ ghsðaÞÞh

s ðtÞDt

Z

a b

¼ a

þ

b

ðg1Þ

ðs þ ghsðaÞÞh

f ðsÞDs þ

Z rðbÞ a

gðaÞ

ðh DarðbÞ f ðtÞÞg s ðtÞDt,

a

f ðtÞDt

f ðtÞDt :

ð15Þ

 g f ðtÞ a Dh g D ðt þ ghÞ

g

ðg1Þ

ðt þ ghaÞh

gðaÞ Dt

ð17Þ

Our main aim is to derive necessary optimality conditions for problem (17).

!

Gðg þ 1Þ

We begin to fix two arbitrary real numbers a and b such that a, b 2 ð0,1. Further, we put g :¼ 1a and n :¼ 1b. Let a function Lðt,u,v,wÞ : Tk  R  R  R-R be given. We consider the problem of minimizing (or maximizing) a functional L : F T -R subject to given boundary conditions: Z b Lðt,ys ðtÞ, a Dah yðtÞ, h Dbb yðtÞÞDt!min, LðyðÞÞ ¼ yðaÞ ¼ A, yðbÞ ¼ B:

Proof. By (9) we can write Z b Z b g f ðtÞa Dah gðtÞDt ¼ f ðtÞða Dh gðt þ ghÞÞD Dt a

Z

g Gðg þ 1Þ

3.2. Necessary optimality conditions

#

b

Theorem 3.2 (Fractional h-summation by parts). Let f and g be real valued functions defined on Tk and T, respectively. Fix 0 o a r1 and put g :¼ 1a. Then, Z b f ðtÞa Dah gðtÞDt

þ

a

¼ hg f ðrðbÞÞgðbÞhg f ðaÞgðaÞ

where the first equality follows from Lemma 2.1. Putting this into (16) we get (15). &

sðtÞ

a

g

f ðtÞa Dh g D ðt þ ghÞDt

Z b Z t g ðg1Þ D ¼ f ðtÞ hg g D ðtÞ þ ðt þ ghsðsÞÞh g ðsÞDs Dt Gðg þ 1Þ a a Z rðbÞ Z b Z b g f ðtÞg D ðtÞDt þ g D ðtÞ ðs þ gh ¼ hg Gðg þ 1Þ a a sðtÞ

gðjh,ihÞf ðjhÞh

j ¼ sðihÞ=h

i ¼ a=h

Z rðbÞ

b=h1 X

kðihÞh

a

sðaÞ

j ¼ b=h1

¼

b

hg f ðrðbÞÞ½gðbÞgðrðbÞÞ Z rðbÞ g þ g D ðtÞh DrðbÞ f ðtghÞDt ¼ hg f ðrðbÞÞ½gðbÞgðrðbÞÞ a Z rðbÞ t ¼ rðbÞ g g þ½gðtÞh DrðbÞ f ðtghÞt ¼ a  g s ðtÞðh DrðbÞ f ðtghÞÞD Dt

j ¼ a=h þ 2

þ    þ kðb2hÞ

Z

where the second equality follows by Lemma 3.1. We proceed to develop the right-hand side of the last equality as follows:

¼ h2 R  C1 ¼ h2 R  ½kðaÞC2 þ kða þ hÞC3 þ    þ kðb2hÞC4  2 b=h1 b=h1 X X ¼ h2 4kðaÞ gðjh,aÞf ðjhÞ þ kða þ hÞ gðjh,a þ hÞf ðjhÞ j ¼ a=h þ 1

ð16Þ Using (8) we get

a

j ¼ a=h

i ¼ a=h

f ðtÞgðaÞDt:

ðg1Þ

b=h1 i1 X X gðt,sÞkðsÞDs Dt ¼ h2 f ðihÞ gðih,jhÞkðjhÞ

a

ðg1Þ

ðt þ ghaÞh

sðtÞÞh f ðsÞDsDt ¼ hg f ðrðbÞÞ½gðbÞgðrðbÞÞ Z rðbÞ g þ g D ðtÞh DrðbÞ f ðtghÞDt,

Direct calculations show that Z

g Gðg þ 1Þ

Definition 3.3. For f 2 F T we define the norm jf s ðtÞj þ max j Da f ðtÞj þ max j Db f ðtÞj: Jf J ¼ max k k a h k h b t2T

t2T

t2T

^ ^ ¼ A and yðbÞ ¼ B is called a A function y^ 2 F T with yðaÞ local minimum for problem (17) provided there exists ^ rLðyÞ for all y 2 F T with y(a)= A and d 4 0 such that LðyÞ ^ o d. y(b)= B and JyyJ Definition 3.4. A function Z 2 F T is called an admissible variation provided Za0 and ZðaÞ ¼ ZðbÞ ¼ 0.

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

Z

From now on we assume that the second-order partial derivatives Luu, Luv, Luw, Lvw, Lvv, and Lww exist and are continuous.

b

¼ a

^ ^ ^ þ h DarðbÞ Lv ½yðtÞ þ a Dbh Lw ½yðtÞ ¼0 Lu ½yðtÞ

ð18Þ

2

holds for all t 2 Tk with operator ½ defined by ½yðsÞ ¼ ðs,ys ðsÞ, a Das yðsÞ, h Dbb yðsÞÞ.

¼ Z

b

a b

¼ a

n

^ Z ðtÞa Dh n ðLw ½yÞðt þ nhÞDt: D

Z

b

a

n

^ ZD ðtÞa Dh ðLw ½yÞðt þ nhÞDt Z rðbÞ a

n ^ ZD ðtÞa D h ðLw ½yÞðt þ nhÞDt n

þ ðZðbÞZðrðbÞÞÞa Dh Z rðbÞ



a ^ Lv ½yðtÞ a Dh ZðtÞDt þ

þ a

a

^ ðLw ½yÞðt þ nhÞjt ¼ rðbÞ n ^ ZðrðbÞÞa Dh ðLw ½yÞðt þ nhÞjt ¼ rðbÞ n ^ ¼ ZðbÞa Dh ðLw ½yÞðt þ nhÞjt ¼ rðbÞ n

b ^ Lw ½yðtÞ h Db ZðtÞDt ¼ 0:

a

a

for the third term in (20). Using (13) it follows that Z b b ^ Lw ½yðtÞ h Db ZðtÞDt Z

b

¼ a

Z

b

¼ a

Z

n D ^ Lw ½yðtÞð h Db ZðtnhÞÞ Dt



b

¼ Z

a b

n Gðn þ 1Þ #

ðtÞÞðhn1Þ D ðsÞDs

ðs þ nhs sðtÞ

Z

ð25Þ

a

By (21) and (25) we may write (20) as Z rðbÞ ^ ^ ^ ½Lu ½yðtÞ þ h DarðbÞ ðLv ½yÞðtÞ þ a Dbh ðLw ½yÞðtÞ Zs ðtÞDt ¼ 0:

ð22Þ

Corollary 3.1 (The h-Euler–Lagrange equation—cf., e.g., Bohner [18], Ferreira and Torres [24]). Let T be the time scale ðhZÞa with the forward jump operator s and the delta derivative D. If y^ is a solution to the problem Z b Lðt,ys ðtÞ,yD ðtÞÞDt!min, yðaÞ ¼ A, yðbÞ ¼ B, LðyðÞÞ ¼ a

s D s D then the equality Lu ðt, y^ ðtÞ, y^ ðtÞÞðLv ðt, y^ ðtÞ, y^ ðtÞÞÞD ¼ 0 2 holds for all t 2 Tk .

We now use Lemma 3.1 to get Z b n D ^ Lw ½yðtÞ h DrðbÞ Z ðtnhÞDt ^ Lw ½yðtÞ hn ZD ðtÞ þ

a

The next result is a direct corollary of Theorem 3.5.

h n D ^ Lw ½yðtÞ h DrðbÞ Z ðtnhÞ

n D ^ Lw ½yðtÞ h DrðbÞ Z ðtnhÞDt Z b nZðbÞ ^ þ ðb þ nhsðtÞÞðhn1Þ Lw ½yðtÞ Dt: Gðn þ1Þ a

b

a

and after inserting in (22), that Z b Z rðbÞ b ^ ^ Lw ½yðtÞ D Z ðtÞ D t ¼ Zs ðtÞa Dbh ðLw ½yÞðtÞ Dt: h b

2

a

Z

^ Zs ðtÞa Dbh ðLw ½yÞðtÞ Dt:

a

¼

a

a

Since the values of Zs ðtÞ are arbitrary for t 2 Tk , the ^ & Euler–Lagrange equation (18) holds along y.

n ðb þ nhsðtÞÞhðn1Þ ZðbÞ Dt Gðn þ1Þ



Z rðbÞ

Since ZðaÞ ¼ ZðbÞ ¼ 0 we obtain, from (23) and (24), that Z b Z rðbÞ n D ^ ^ Lw ½yðtÞ Zs ðtÞa Dbh ðLw ½yÞðtÞ Dt, h DrðbÞ Z ðtÞDt ¼ 

Using Theorem 3.2 and the fact that ZðaÞ ¼ ZðbÞ ¼ 0, we get Z b Z rðbÞ a ^ ^ Lv ½yðtÞ ðh DarðbÞ ðLv ½yÞðtÞÞ Zs ðtÞDt ð21Þ a Dh ZðtÞDt ¼

a

^ ðLw ½yÞðt þ nhÞjt ¼ a 

ð24Þ

ð20Þ

a

nhÞÞD Dt

n

ZðaÞa Dh

b

nhÞtt ¼¼ arðbÞ

n ^ Zs ðtÞða D h ðLw ½yÞðt þ

þ ZðbÞa Dh

a

Z

^ ðLw ½yÞðt þ nhÞjt ¼ rðbÞ

n ^ ¼ ½ZðtÞa Dh ðLw ½yÞðt þ

ð19Þ

a

ð23Þ

We apply again the time scale integration by parts formula (Lemma 2.1), this time to (23), to obtain

This function has a minimum at e ¼ 0, so we must have F0 ð0Þ ¼ 0, i.e., Z b b a ^ ^ ^ ½Lu ½yðtÞ Zs ðtÞ þLv ½yðtÞ a Dh ZðtÞ þ Lw ½yðtÞh Db ZðtÞDt ¼ 0,

b

Dt

a

a

Z

Z

^ ZD ðtÞDt þ hn Lw ½yðtÞ Gðn þ1Þ

Z t ^ ZD ðtÞ ðt þ nhsðsÞÞhðn1Þ Lw ½yðsÞ Ds Dt

b

¼

which we may write equivalently as Z rðbÞ ^ ^ Zs ðtÞjt ¼ rðbÞ þ Lu ½yðtÞ Zs ðtÞDt hLu ½yðtÞ

ðs þ nhs

^ Lw ½yðtÞ

#

ðtÞÞðhn1Þ D ðsÞDs

sðtÞ

Z

^ Proof. Suppose that yðÞ is a local minimum of LðÞ. Let ZðÞ be an arbitrarily fixed admissible variation and define a function F : ðd=JZðÞJ, d=JZðÞJÞ-R by ^ þ eZðÞÞ: FðeÞ ¼ LðyðÞ

b

a

Z

a

Theorem 3.5 (The h-fractional Euler–Lagrange equation for problem (17)). If y^ 2 F T is a local minimum for problem (17), then the equality

n

^ ZD ðtÞDt þ hn Lw ½yðtÞ Gðn þ1Þ " Z

Z rðbÞ

3.2.1. First order optimality condition Next theorem gives a first order necessary condition for problem (17), i.e., an Euler–Lagrange type equation for the fractional h-difference setting.

519

Proof. Choose a ¼ 1 and a L that does not depend on w in Theorem 3.5. & Remark 3.1. If we take h= 1 in Corollary 3.1 we have that

Dt

s ^ ^ Lu ðt, y^ ðtÞ, DyðtÞÞ ¼0 DLv ðt, y^ s ðtÞ, DyðtÞÞ

520

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524 2

2

holds for all t 2 Tk . This equation is usually called the discrete Euler–Lagrange equation, and can be found, e.g., in [29, Chapter 8].

Let t 2 Tk be arbitrary, and choose Z : T-R given by ( h if t ¼ sðtÞ; ZðtÞ ¼ 0 otherwise:

3.2.2. Natural boundary conditions If the initial condition y(a)= A is not present in problem (17) (i.e., y(a) is free), besides the h-fractional Euler– Lagrange equation (18) the following supplementary condition must be fulfilled: Z b g ðg1Þ ^ ^ hg Lv ½yðaÞ þ ðt þ ghaÞh Lv ½yðtÞ Dt Gðg þ1Þ a ! Z

It follows that ZðaÞ ¼ ZðbÞ ¼ 0, i.e., Z is an admissible variation. Using (9) we get Z b ^ ^ ½Luu ½yðtÞð Zs ðtÞÞ2 þ 2Luv ½yðtÞ Zs ðtÞa Dah ZðtÞ

b

 sðaÞ

ðg1Þ

ðt þ ghsðaÞÞh

^ ^ Lv ½yðtÞ Dt þ Lw ½yðaÞ ¼ 0:

ð26Þ

Similarly, if y(b)= B is not present in (17) (y(b) is free), the extra condition ^ rðbÞÞ þ hg Lv ½yð ^ rðbÞÞhn Lw ½yð ^ rðbÞÞ hLu ½yð Z b n ^ þ ðbþ nhsðtÞÞðhn1Þ Lw ½yðtÞ Dt Gðn þ 1Þ a ! Z rðbÞ ^ ðrðbÞ þ nhsðtÞÞðhn1Þ Lw ½yðtÞ Dt ¼ 0 

ð27Þ

is added to Theorem 3.5. We leave the proof of the natural boundary conditions (26) and (27) to the reader. We just note here that the first term in (27) arises from the first term of the left-hand side of (20). 3.2.3. Second order optimality condition We now obtain a second order necessary condition for problem (17), i.e., we prove a Legendre optimality type condition for the fractional h-difference setting. Theorem 3.6 (The h-fractional Legendre necessary condition). If y^ 2 F T is a local minimum for problem (17), then the inequality

^ tÞ þ 2hg þ 2 Luv ½yð ^ tÞ þ hg þ 1 Lvv ½yð ^ tÞ ¼ h3 Luu ½yð  Z b g ^ þ Lvv ½yðtÞ hg ZD ðtÞ þ Gðg þ1Þ sðtÞ 2 Z t ðg1Þ D ðt þ ghsðsÞÞh Z ðsÞDs Dt: a

Observe that ^ sðtÞÞ þ h2g þ 1 ðg1Þ2 Lvv ½yð

Z

t

s

g t

sðtÞ

¼h

Z ðsÞDs

Z sðtÞ

ðg1Þ D

ðt þ ghsðsÞÞh

ðg1Þ D ðt þ ghsðsÞÞh Z ðsÞDs

Gðg þ1Þ Z

a

b

^ ^ ^ ¼ ðt, y^ ðtÞ, a Dh yðtÞ, holds for all t 2 T , where ½yðtÞ h Db yðtÞÞ. Proof. By the hypothesis of the theorem, and letting F be as in (19), we have as necessary optimality condition that 00 F ð0Þ Z 0 for an arbitrary admissible variation ZðÞ. 00 Inequality F ð0Þ Z 0 is equivalent to Z b ^ ^ Zs ðtÞa Dah ZðtÞ ½Luu ½yðtÞð Zs ðtÞÞ2 þ 2Luv ½yðtÞ a

a 2 ^ ^ Zs ðtÞh Dbb ZðtÞ þLvv ½yðtÞð þ2Luw ½yðtÞ a Dh ZðtÞÞ b b a 2 ^ þ2Lvw ½yðtÞ a Dh ZðtÞh Db ZðtÞ þ Lww ðtÞðh Db ZðtÞÞ Dt Z 0:

ð29Þ

g Gðg þ 1Þ

ðg1Þ D

ðt þ ghsðsÞÞh

a

Z ðsÞDs

a

g

ðg1Þ

Gðg þ1Þ

½ðt þ ghsðtÞÞh

ðg1Þ

ðt þ ghsðsðtÞÞÞh



    3   2 tt tt tt tt G þ g1 G þ g1  þ g1 g 7 gh 6 h h h 7 6 h     ¼ 5 tt tt Gðg þ1Þ 4

ð28Þ k2



Let t 2 ½s2 ðtÞ, rðbÞ \ ðhZÞa . Since

þ



s2 ðtÞ

^ Lvv ½yðtÞ

a

¼

2 nð1nÞ ^ ðt þ nhsðsÞÞðhn2Þ Ds þ hg Lvv ½yðtÞ Gðn þ 1Þ  2 Z b gðg1Þ ðg2Þ ^ ðs þ ghsðsðtÞÞÞh þ h3 Lvv ½yðsÞ Ds Z0 Gðg þ 1Þ sðsðtÞÞ

b

t

Gðg þ1Þ

^ ^ þ h2n ðn1Þ2 Lww ½yðtÞ þ2hn þ g ðn1ÞLvw ½yðtÞ Z t 2n 3 ^ sðtÞÞ þ ^ h Lww ½yðsÞ þh Lww ½yð

Z

2 ðg1Þ D ðt þ ghsðsÞÞh Z ðsÞDs Dt a  Z b g ^ ¼ Lvv ½yðtÞ hg ZD ðtÞ þ Gðg þ 1Þ sðtÞ 2 Z t ðg1Þ D ðt þ ghsðsÞÞh Z ðsÞDs Dt: Z

g

^ ^ ^ h2 Luu ½yðtÞ þ2hg þ 1 Luv ½yðtÞ þ 2hn þ 1 ðn1ÞLuw ½yðtÞ 2 2g ^ sðtÞÞ þ 2hn þ g ðg1ÞLvw ½yð ^ sðtÞÞ þh ðg1Þ Lvv ½yð



a 2 ^ þ Lvv ½yðtÞð a Dh ZðtÞÞ Dt Z bh ^ ^ Luu ½yðtÞð Zs ðtÞÞ2 þ 2Luv ½yðtÞ Zs ðtÞ ¼ a   Z t g ðg1Þ D ðt þ ghsðsÞÞh Z ðsÞDs hg ZD ðtÞ þ Gðg þ1Þ a  g ^ hg ZD ðtÞ þ þ Lvv ½yðtÞ Gðg þ 1Þ 2 i Z t ðg1Þ D ðt þ ghsðsÞÞh Z ðsÞDs Dt a

a

a

a

h ¼ h2

gðg1Þ Gðg þ 1Þ

ðg2Þ

ðt þ ghsðsðtÞÞÞh

G

h

,

ð30Þ

we conclude that  2 Z b Z t g ðg1Þ D ^ Lvv ½yðtÞ ðt þ ghsðsÞÞh Z ðsÞDs Dt Gðg þ 1Þ a s2 ðtÞ  2 Z b gðg1Þ ðg2Þ ^ ðt þ ghs2 ðtÞÞh Lvv ½yðtÞ h2 Dt: ¼ Gðg þ 1Þ s2 ðtÞ n D Note that we can write t Dbb ZðtÞ ¼ h DrðbÞ Z ðtnhÞ because ZðbÞ ¼ 0. It is not difficult to see that the following

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

equality holds: Z b ^ 2Luw ½yðtÞ Zs ðtÞh Dbb ZðtÞDt a Z b n D ^ ¼ 2Luw ½yðtÞ Zs ðtÞh DrðbÞ Z ðtnhÞDt ¼ 2h

a 2þn

LðyðÞÞ ¼



b

Proof. Choose a ¼ 1 and a Lagrangian L that does not depend on w. Then, g ¼ 0 and the result follows immediately from Theorem 3.6. &

b

Remark 3.2. When h goes to zero we have sðtÞ ¼ t and inequality (32) coincides with Legendre’s classical neces^ sary optimality condition Lvv ½yðtÞ Z 0 (cf., e.g., [50]). #)

ðtÞÞðhn1Þ D ðsÞDs

ðs þ nhs

sðtÞ

Z

4. Examples

Dt In this section we present some illustrative examples. Example 4.1. Let us consider the following problem: Z 2 1 1  3=4 LðyÞ ¼ Dt!min, yð0Þ ¼ 0, yð1Þ ¼ 1: 0 Dh yðtÞ 2 0 ð33Þ

b 2 ^ Lww ½yðtÞð h Db ZðtÞÞ Dt

Z sðsðtÞÞ ¼

" ^ Lww ½yðtÞ hn ZD ðtÞþ

a

Z t ¼

" ^ Lww ½yðtÞ

a

n Gðn þ 1Þ

Z

b

sðtÞ

n Gðn þ 1Þ

Z

b

sðtÞ

#2 ðs þ nhsðtÞÞhðn1Þ ZD ðsÞDs

Dt

#2 ðs þ nhsðtÞÞhðn1Þ ZD ðsÞDs

Dt

^ tÞðhn nhn Þ2 þ h2n þ 1 L ½yð

¼

þ hLww Z t ^ Lww ½yðtÞ h a

sðtÞÞðhn1Þ

i2

^ sðtÞÞ ww ½yð n n ðt þ nhsðtÞÞhðn1Þ ðsðtÞ þ nh Gðn þ 1Þ

^ tÞðhn nhn Þ2 þ h2n þ 1 Lww ½yð ^ sðtÞÞ: þ hLww ½yð

We consider (33) with different values of h. Numerical results show that when h tends to zero the h-fractional Euler–Lagrange extremal tends to the fractional continuous extremal: when h-0 (33) tends to the fractional continuous variational problem in the Riemann–Liouville sense studied in [1, Example 1], with solution given by Z 1 t dx yðtÞ ¼ : ð34Þ 2 0 ½ð1xÞðtxÞ1=4 This is illustrated in Fig. 1. In this example for each value of h there is a unique h-fractional Euler–Lagrange extremal, solution of (18), which always verifies the h-fractional Legendre necessary condition (28).

Similarly as we did in (30), we can prove that

n fðt þ nhsðtÞÞðhn1Þ ðsðtÞ þ nhsðtÞÞðhn1Þ g h Gðn þ 1Þ nð1nÞ ðt þ nhsðtÞÞðhn2Þ : ¼ h2 Gðn þ 1Þ Thus, we have that inequality (29) is equivalent to  ^ ^ ^ h h2 Luu ½yðtÞþ 2hg þ 1 Luv ½yðtÞþ hg Lvv ½yðtÞ þLvv ðsðtÞÞðghg hg Þ2

1.0 0.9

 2 gðg1Þ ðg2Þ ðs þ ghsðsðtÞÞÞh þ h3 Lvv ðsÞ Ds Gðg þ 1Þ sðsðtÞÞ nþ1 g þ n ^ ^ þ2h Luw ½yðtÞð n1Þþ 2h ðn1ÞLvw ½yðtÞ Z

b

þ2hg þ n ðg1ÞL

2n

0.8 0.7

2

y˜ (t)

^ sðtÞÞ þ h Lww ½yðtÞð1 nÞ  Z t n ð1nÞ 3 ^ sðtÞÞ þ ^ ðt þ nh þh Lww ½yð h Lww ½yðsÞ Gðn þ 1Þ a vw ð

2n

sðsÞÞn2

2



Ds Z 0:

ð32Þ

s D ^ ¼ ðt, y^ ðtÞ, y^ ðtÞÞ. holds for all t 2 T , where ½yðtÞ

Finally, we have that a

yðbÞ ¼ B,

k2

^ tÞ þ 2hg þ n þ 1 ðg1ÞLvw ½yð ^ sðtÞÞ: ¼ 2hg þ n þ 1 ðn1ÞLvw ½yð Z

yðaÞ ¼ A,

^ ^ ^ ^ sðtÞÞ Z 0 þ 2hLuv ½yðtÞ þ Lvv ½yðtÞ þ Lvv ½yð h2 Luu ½yðtÞ

 Z b g ^ ¼ 2 Lvw ½yðtÞ hg ZD ðtÞ þ Gðg þ 1Þ a  Z t ðg1Þ D ðt þ ghsðsÞÞh Z ðsÞDs  "a Z

n Gðn þ1Þ

Lðt,ys ðtÞ,yD ðtÞÞDt!min,

then the inequality

^ tÞðn1Þ: Luw ½yð

hn ZD ðtÞ þ

b a

Moreover, Z b b a ^ 2Lvw ½yðtÞ a Dh ZðtÞh Db ZðtÞDt a

Z

521

ð31Þ

Because h4 0, (31) is equivalent to (28). The theorem is proved. &

0.6 0.5 0.4 0.3 0.2 0.1

The next result is a simple corollary of Theorem 3.6. Corollary 3.2 (The h-Legendre necessary condition—cf. Result 1.3 of Bohner [18]). Let T be the time scale ðhZÞa with the forward jump operator s and the delta derivative D. If y^ is a solution to the problem

0

0

1 t

~ Fig. 1. Extremal yðtÞ for problem of Example 4.1 with different values of 1 h: h = 0.50 ðÞ; h = 0.125 (+ ); h = 0.0625 (); h ¼ 30 ðÞ. The continuous line represents function (34).

522

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

Example 4.2. Let us consider the following problem: LðyÞ ¼

Z

1

0



1 ð0 Dah yðtÞÞ2 ys ðtÞ Dt!min, 2

yð0Þ ¼ 0,

0.35

yð1Þ ¼ 0:

0.30

We begin by considering problem (35) with a fixed value for a and different values of h. The extremals y~ are obtained using our Euler–Lagrange equation (18). As in Example 4.1 the numerical results show that when h tends to zero the extremal of the problem tends to the extremal of the corresponding continuous fractional problem of the calculus of variations in the Riemann– Liouville sense. More precisely, when h approximates zero problem (35) tends to the fractional continuous problem studied in [2, Example 2]. For a ¼ 1 and h-0 the extremal of (35) is given by yðtÞ ¼ 12 tð1tÞ, which coincides with the extremal of the classical problem of the calculus of variations  Z 1 1 0 2 y ðtÞ yðtÞ dt!min, yð0Þ ¼ 0, yð1Þ ¼ 0: LðyÞ ¼ 2 0

0.25

This is illustrated in Fig. 2 for h ¼ 1=2i , i= 1, 2, 3, 4. In this example, for each value of a and h, we only have one extremal (we only have one solution to (18) for each a and h). Our Legendre condition (28) is always verified along the extremals. Fig. 3 shows the extremals of problem (35) 1 for a fixed value of h ðh ¼ 20 Þ and different values of a. The numerical results show that when a tends to one the extremal tends to the solution of the classical (integer order) discrete-time problem. Our last example shows that the h-fractional Legendre necessary optimality condition can be a very useful tool. In Example 4.3 we consider a problem for which the hfractional Euler–Lagrange equation gives several candidates but just a few of them verify the Legendre condition (28). Example 4.3. Let us consider the following problem: Z b LðyÞ ¼ ða Dah yðtÞÞ3 þ yðh Dab yðtÞÞ2 Dt!min, a

yðaÞ ¼ 0,

yðbÞ ¼ 1:

ð36Þ

y˜ (t)

ð35Þ

0.20 0.15 0.10 0.05 0

0

1 t

~ Fig. 3. Extremal yðtÞ for (35) with h = 0.05 and different values of a: a ¼ 0:70 ðÞ; a ¼ 0:75 ðÞ; a ¼ 0:95 (+ ); a ¼ 0:99 (). The continuous line is yðtÞ ¼ 12 tð1tÞ.

For a ¼ 0:8, b ¼ 0:5, h= 0.25, a= 0, b= 1, and y ¼ 1, problem (36) has eight different Euler–Lagrange extremals. As we can see in Table 1 only two of the candidates verify the Legendre condition. To determine the best candidate we compare the values of the functional L along the two good candidates. The extremal we are looking for is given by the candidate number five in Table 1. For problem (36) with a ¼ 0:3, h= 0.1, a = 0, b= 0.5, and

y ¼ 0, we obtain the results of Table 2: there exist 16 Euler–Lagrange extremals but only one satisfy the fractional Legendre condition. The extremal we are looking for is given by the candidate number six in Table 2. The numerical results show that the solutions to our discrete-time fractional variational problems converge to the classical discrete-time solutions when the fractional order of the discrete-derivatives tend to integer values, and to the fractional Riemann–Liouville continuous-time solutions when h tends to zero. 5. Conclusion

y˜ (t)

0.15

0.10

0.05

0

0

1 t

~ Fig. 2. Extremal yðtÞ for problem (35) with a ¼ 1 and different values of h: h = 0.5 ðÞ; h = 0.25 ðÞ; h = 0.125 (+ ); h = 0.0625 ().

The discrete fractional calculus is a recent subject under strong current development due to its importance as a modeling tool of real phenomena. In this work we introduce a new fractional difference variational calculus in the time-scale ðhZÞa ,h 4 0 and a a real number, for Lagrangians depending on left and right discrete-time fractional derivatives. Our objective was to introduce the concept of left and right fractional sum/difference (cf. Definition 2.8) and to develop the theory of fractional difference calculus. An Euler–Lagrange type Eq. (18), fractional natural boundary conditions (26) and (27), and a second order Legendre type necessary optimality condition (28), were obtained. The results are based on a new discrete fractional summation by parts formula (15) for ðhZÞa . Obtained first and second order necessary optimality conditions were implemented computationally

N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

523

Table 1 There exist eight Euler–Lagrange extremals for problem (36) with a ¼ 0:8, b ¼ 0:5, h = 0.25, a= 0, b= 1, and y ¼ 1, but only two of them satisfy the fractional Legendre condition (28). #

1 2 3 4 5 6 7 8

y~

  1 4

 0.5511786 0.2669091  2.6745703 0.5789976 1.0306820 0.5087946 4.0583690  1.7436106

y~

  1 2

0.0515282 0.4878808 0.5599360 1.0701515 1.8920322  0.1861431  1.0299054  3.1898449

y~

  3 4

0.5133134 0.7151924  2.6730125 0.1840377 2.7429222 0.4489196  5.0030989  0.8850511

~ LðyÞ

Legendre condition (28)

9.3035911 2.0084203 698.4443232 12.5174960  32.7189756 10.6730959 2451.7637948 238.6120299

Not verified Verified Not verified Not verified Verified Not verified Not verified Not verified

Table 2 There exist 16 Euler–Lagrange extremals for problem (36) with a ¼ 0:3, h = 0.1, a= 0, b= 0.5, and y ¼ 0, but only one (candidate #6) satisfy the fractional Legendre condition (28). #

~ yð0:1Þ

~ yð0:2Þ

~ yð0:3Þ

~ yð0:4Þ

~ LðyÞ

(28)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

 0.305570704  0.427934654 0.284152257  0.277642565 0.387074742 0.259846344  0.375094681 0.343327771 0.297792192 0.41283304  0.321401682 0.330157414  0.459640837  0.359429958 0.477760586  0.541587541

 0.428093486  0.599520948  0.227595659 0.222381632  0.310032839 0.364035314 0.300437245 0.480989769 0.417196073 0.578364133 0.257431098  0.264444122 0.368155651  0.50354835  0.382668914  0.758744525

0.223708338 0.313290997 0.318847274 0.386666793 0.434336603 0.463222456 0.522386246 0.61204299  0.218013689  0.302235104  0.360644857  0.459803086  0.515763025  0.640748011  0.66536683  0.965476394

0.480549114  0.661831134 0.531827387 0.555841555  0.482903047 0.597907505  0.419053781  0.280908953 0.460556635  0.649232892 0.400971272 0.368850105  0.860276767 0.294083676  0.956478654  1.246195157

12.25396166 156.2317667 8.669645848 6.993518478 110.7912605 5.104389191 93.95316858 69.23497954 14.12227593 157.8272685 19.87468886 24.84475504 224.9964788 34.43515839 263.3075289 392.9592508

No No No No No Yes No No No No No No No No No No

in the computer algebra systems Maple and Maxima. Our numerical results show that: 1. the solutions of our fractional problems converge to the classical discrete-time solutions in ðhZÞa when the fractional order of the discrete-derivatives tend to integer values; 2. the solutions of the considered fractional problems converge to the fractional Riemann–Liouville continuous solutions when h-0; 3. there are cases for which the fractional Euler–Lagrange equation gives only one candidate that does not verify the obtained Legendre condition (so the problem at hands does not have a minimum); 4. there are cases for which the Euler–Lagrange equation gives only one candidate that verify the Legendre condition (so the extremal is a candidate for minimizer, not for maximizer); 5. there are cases for which the Euler–Lagrange equation gives us several candidates and just a few of them verify the Legendre condition. We can say that the obtained Legendre condition can be a very practical tool to conclude when a candidate identified via the Euler–Lagrange equation is really a solution of the fractional variational problem. It is worth to mention

that a fractional Legendre condition for the continuous fractional variational calculus is still an open question. Undoubtedly, much remains to be done in the development of the theory of discrete fractional calculus of variations in ðhZÞa here initiated. Moreover, we trust that the present work will initiate research not only in the area of the discrete-time fractional calculus of variations but also in solving fractional difference equations containing left and right fractional differences. One of the subjects that deserves special attention is the question of existence of solutions to the discrete fractional Euler–Lagrange equations. Note that the obtained fractional equation (18) involves both the left and the right discrete fractional derivatives. Other interesting directions of research consist to study optimality conditions for more general variable endpoint variational problems [28,34,35]; isoperimetric problems [5,6]; higher-order problems of the calculus of variations [12,24,38]; to obtain fractional sufficient optimality conditions of Jacobi type and a version of Noether’s theorem [17,21,25,27] for discrete-time fractional variational problems; direct methods of optimization for absolute extrema [19,36,49]; to generalize our fractional first and second order optimality conditions for a fractional Lagrangian possessing delay terms [14,37]; and to generalize the results from ðhZÞa to an arbitrary time scale T.

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N.R.O. Bastos et al. / Signal Processing 91 (2011) 513–524

Acknowledgments This work is part of the first author’s PhD project carried out at the University of Aveiro under the framework of the Doctoral Programme Mathematics and Applications of Universities of Aveiro and Minho. The financial support of the Polytechnic Institute of Viseu and The Portuguese Foundation for Science and Technology (FCT), through the ‘‘Programa de apoio a formac- a~ o avanc- ada de docentes do Ensino Superior Polite´cnico’’, PhD fellowship SFRH/PROTEC/49730/2009, is here gratefully acknowledged. The second author was supported by FCT through the PhD fellowship SFRH/BD/ 39816/2007; the third author by FCT through the R&D unit Centre for Research on Optimization and Control (CEOC) and the project UTAustin/MAT/0057/2008.

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