# Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

Authors: Jun Jiang; Yuqiang Feng; Shougui Li
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.


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## Body

## 1. Introduction

Since the introduction of fractional calculus of variations by Riewe [1], fractional calculus has been a subject of interest not only among mathematicians, but also among fluid mechanics, electricity and finance specialists, chemical physicists, biomedical engineering specialists, and control theory specialists.Considerable progress has been made to determine necessary and sufficient conditions that any extremal for the variational functional with fractional calculus must satisfy in recent years. R. Almeida [2] provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Almeida established the fractional Euler-Lagrange equations for the fundamental problem and when in presence of an integral constraint and Almeida obtained a Legendre condition. In [3] Almeida studied certain problems of calculus of variations that are dependent upon a Lagrange function on a Caputo-type fractional derivative; sufficient and necessary conditions of the first- and second-order are presented. In [4] Zhang Jianke, Ma Xiaojue, and Li Lifeng studied the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function depending on a Caputo-Fabrizio fractional derivative. In [5] Almeida et al. obtained necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative and indefinite integral. There has been a significant development in ordinary and partial fractional differential equations in recent years [6–9].D. Tavares et al. in [10] presented two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and presented a new variational problem subject to holonomic constraint.Noether’s symmetry, namely, the invariance of Hamilton action under the infinitesimal transformations, is put forward for the first time by Noether [11]. In [12] Frederico et al. obtained a generalization of the Noether theorem for Lagrangians depending on mixed classical and Caputo derivatives that can be used to obtain constants of motion for dissipative systems. In [13] the Noether theorem and its inverse theorem for the nonlinear dynamical systems with nonstandard Lagrangians are studied. In [14] a variational principle for Lagrangian densities containing derivatives of real order was formulated and the invariance of this principle is studied in two characteristic cases. In [15] Yan B. et al. studied Noether’s symmetries and conserved quantities of the Birkhoffian systems in terms of fractional derivatives of variable order.In this paper, we extend the study to variational problems involving partial fractional derivatives. Our aim is to obtain the necessary and sufficient conditions for the minimizer. Moreover, we will establish Noether’s theorem for these problems.The paper is organized as follows. In Section2, the basic definitions and notations are given. In Sections 3.1 and 3.2, the first-order necessary condition and the second-order necessary condition for the minimizer are established. Then, the variational problem subject to an integral constraint is investigated in Section 3.3 and the isoperimetric problem is discussed in Section 3.4. In Section 3.5, Noether’s theorem for this system is proved. In Section 4, as applications of our results, some examples are presented.


## 2. Preliminaries

In this section, we will recall some basic concepts and preliminary results on fractional calculus, needed in the sequel. To fix notation, in the followingD=[a,b]×[c,d] and ∂D is the boundary of D.Definition 1 (Riemann-Liouville fractional integrals, [12]).
The left and right Riemann-Liouville fractional integrals of orderα for the function f(x) are defined, respectively, as (1)Ia+αft≔1Γα∫att-τα-1fτdτIb-αft≔1Γα∫tbτ-tα-1fτdτ where Γ is the gamma function.Definition 2 (Riemann-Liouville fractional derivatives, [12]).
The left and right Riemann-Liouville fractional derivatives of orderα for the function f(x) are defined, respectively, as (2)Da+αft≔1Γ1-αddt∫att-τ-αfτdτDb-αft≔1Γ1-αddt∫tbτ-t-αfτdτRemark 3.
The Riemann-Liouville fractional derivative of a constantc need not to be zero.Definition 4 (Caputo fractional derivatives, [12]).
The left Caputo fractional derivative off(x) of order α∈0,1 is defined by (3)Da+αCft≔1Γ1-αddt∫at1t-ταfτ-fadτ If f is of class C1, then we have the equivalent form (4)Da+αCft≔1Γ1-α∫at1t-ταf′τdτ while the right Caputo fractional derivative of f(x) of order α∈0,1 is given by (5)Db-αCft≔1Γ1-α∫tb1τ-tαf′τdτRemark 5.
The Caputo derivative exhibits an important feature: the derivative of a constant is zero.Definition 6 (partial Riemann-Liouville integrals, [9]).
Letα∈0,1 and u∈L1(D). The left and right partial Riemann-Liouville integrals of order α∈0,1 of u(x,y) with respect to x are defined, respectively, by the expression (6)Ia,xαux,y=1Γα∫axx-sα-1us,ydsIx,bαux,y=1Γα∫xbs-xα-1us,yds for almost all (x,y)∈D. Analogously, we define the integrals (7)Ic,yαux,y=1Γα∫cyy-tα-1ux,tdtIy,dαux,y=1Γα∫ydt-yα-1ux,tdt for almost all (x,y)∈D.Definition 7 (partial Riemann-Liouville derivatives, [9]).
Letα∈0,1 and u∈L1(D). The left and right partial Riemann-Liouville derivatives of order α∈0,1 of u(x,y) with respect to x are defined, respectively, by the expression (8)Da,xαux,y=∂∂xIa,x1-αux,yDx,bαux,y=∂∂xIx,b1-αux,y for almost all (x,y)∈D. Analogously, we define the derivatives (9)Dc,yαux,y=∂∂yIc,y1-αux,yDy,dαux,y=∂∂yIy,d1-αux,y for almost all (x,y)∈D.Definition 8 (partial Caputo fractional derivative, [9]).
Letα∈0,1 and u∈L1(D). The left and right partial Caputo fractional derivatives of order α∈0,1 of u(x,y) with respect to x are defined, respectively, by the expression (10)Da,xαCux,y=Ia,x1-α∂∂xux,yDx,bαCux,y=Ix,b1-α∂∂xux,y for almost all (x,y)∈D. Analogously, we define the derivatives (11)Dc,yαCux,y=Ic,y1-α∂∂yux,yDx,bαCux,y=Ix,b1-α∂∂xux,y for almost all (x,y)∈D.Theorem 9.
Letf(x,y) be a continuous function and g(x,y) be of class C1; then (12)∬Dfx,y·Da,xαCgx,ydxdy=∬D∂∂xgx,y·Ix,b1-αfx,ydxdy-∬Dgx,y·Dx,bαfx,ydxdy
and(13)∬Dfx,y·Dc,yαCgx,ydxdy=∬D∂∂ygx,y·Iy,d1-αfx,ydxdy-∬Dgx,y·Dy,dαfx,ydxdyProof.
(14) ∬ D f x , y · D a , x α   C g x , y d x d y = ∫ c d ∫ a b f x , y · 1 Γ 1 - α ∫ a x x - s - α ∂ ∂ s g s , y d s d x d y = ∫ c d ∫ a b f x , y · 1 Γ 1 - α ∫ a x x - s - α ∂ ∂ s g s , y d s d x d y = ∫ c d ∫ a b ∂ ∂ s g s , y · 1 Γ 1 - α ∫ s b x - s - α f x , y d x d s d y = ∫ c d ∫ a b ∂ ∂ x g x , y · 1 Γ 1 - α ∫ x b s - x - α f s , y d s d x d y = ∫ c d ∫ a b ∂ ∂ x g x , y · 1 Γ 1 - α ∫ x b s - x - α f s , y d s d x d y = ∬ D ∂ ∂ x g x , y Γ 1 - α ∫ x b s - x - α f s , y d s d x d y - ∬ D g x , y Γ 1 - α ∂ ∂ x ∫ x b s - x - α f s , y d s d x d y = ∬ D ∂ ∂ x g x , y · I x , b 1 - α f x , y d x d y - ∬ D g x , y · D x , b α f x , y d x d y Analogously, we obtain (15)∬Dfx,y·Dc,yαCgx,ydxdy=∬D∂∂ygx,y·Iy,d1-αfx,ydxdy-∬Dgx,y·Dy,dαfx,ydxdyTheorem 10 (see [16]).
Iff(x,y)∈C1(D) satisfies (16)∬Dfx,yηx,ydxdy=0 for all η(x,y)∈C1(D) with η(x,y)∂D=0, then f(x,y)≡0 on D.


## 3. The Variational Problem

The aim of this section is to study problems of fractional calculus of variations, where the integral functional depends on the partial Caputo fractional derivative. Givenu(x,y)∈C1(D), we define the functional (17)Jux,y=∬DFx,y,ux,y,uxx,y,uyx,y,Da,xαCux,y,Dc,yαCux,ydxdywith the following assumptions:(i)F:D×R5→R is a continuous function, such that ∂3F,∂4F,∂5F,∂6F, and ∂7F exist and are continuous.(ii) Given anyu(x,y)∈C1(D), Dx,bαC∂6F and Dy,dαC∂7F are continuous.For simplification, we consider the operator(18)ux,y=x,y,ux,y,uxx,y,uyx,yDa,xαCux,y,Dc,yαCux,y∂ihx1,x2,…,xn=∂h∂xix1,x2,…,xn∂ij2hx1,x2,…,xn=∂2h∂xi∂xjx1,x2,…,xn


### 3.1. The Fundamental Problem

Theorem 11.
Suppose thatu∗(x,y) is a local minimizer for J as in (17), defined on (19)U=ux,y∈C1D∣ux,c=φx,ua,y=ψy,x∈a,b,y∈c,d where φ(x)∈AC([a,b]) and ψ(y)∈AC([c,d]). Then u∗(x,y) is a solution to the equation (20)∂3Fu∗x,y-∂∂x∂4Fu∗x,y-∂∂y∂5Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂7Fu∗x,y=0for almost all (x,y)∈D.Proof.
Letu(x,y)+εη(x,y) be a variation of u(x,y), with ε≪1, ηx,y∈C1D∩U, and η(x,y)∂D=0. We define the function h in a neighborhood of zero by the expression (21)hε=Ju∗+εη
Sinceu∗ is a minimizer of J, then ε=0 is a minimizer of h, and so h′(0)=0. Differentiating h at ε=0 and using Theorem 9, we obtain(22)h′0=∬D∂3Fu∗x,y·ηx,y+∂4Fu∗x,y·∂xηx,y+∂5Fu∗x,y·∂yηx,y+∂6Fu∗x,y·Da,xαCηx,y+∂7Fu∗x,y·Dc,yαCηx,ydxdy⇒h′0=∬D∂3Fu∗x,y-∂∂x∂4Fu∗x,y-∂∂y∂5Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂7Fu∗x,yηx,ydxdy+∬D∂∂x∂4Fu∗x,y+Ix,b1-α∂6Fu∗x,yηx,ydxdy+∬D∂∂y∂5Fu∗x,y+Iy,d1-α∂7Fu∗x,yηx,ydxdy⇒h′0=∬D∂3Fu∗x,y-∂∂x∂4Fu∗x,y-∂∂y∂5Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂7Fu∗x,yηx,ydxdy-∮∂D∂5Fu∗x,y+Iy,d1-α∂7Fu∗x,yηx,ydx+∮∂D∂4Fu∗x,y+Ix,b1-α∂6Fu∗x,yηx,ydy⇒h′0=0.
Sinceη(x,y)∂D=0 and η(x,y) is arbitrary elsewhere, we conclude that (23)∂3Fu∗x,y-∂∂x∂4Fu∗x,y-∂∂y∂5Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂7Fu∗x,y=0 for almost all (x,y)∈D.Remark 12.
Equation (20) is called the Euler-Lagrange equation associated with the functional J. Solutions of this equation are called extremals.Remark 13.
If we define functional(24)Jux,y=∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy then we obtain the Euler-Lagrange equation 20′∂3Fu∗x,y-Dx,bα∂4Fu∗x,y-Dy,dα∂5Fu∗x,y=0for almost all (x,y)∈D.


### 3.2. The Legendre Condition

In [17], a second-order necessary condition had been proved for functionals involving Riemann-Liouville fractional derivatives.In this section, we give a second-order necessary condition, usually called Legendre condition, for functionals involving Riemann-Liouville partial fractional derivatives.We introduce the functional(25)Jux,y=∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdywith the same assumptions on F as in Section 3.1.Theorem 14.
Suppose thatu∗ is a local minimizer for J in U. If ∂ij2F exists and is continuous for i,j∈{3,4,5}, then u∗ satisfies (26)∂44Fu∗x,y≥0(27)∂55Fu∗x,y≥0for almost all (x,y)∈D.Proof.
Letu∗(x,y)+εη(x,y) be a variation of u∗(x,y), with ε≪1, ηx,y∈C1D∩U, and η(x,y)∂D=0.
We define the functionh in a neighborhood of zero by the expression h(ε)=J(u∗+εη). Then, we have h′′(0)≥0; that is, (28)∬D∂33Fu∗x,y·η2x,y+2∂34Fu∗x,y·ηx,y·Da,xαCηx,y+2∂35Fu∗x,y·ηx,y·Dc,yαCηx,y+∂44Fu∗x,y·Da,xαCηx,y2+2∂45Fu∗x,y·Da,xαCηx,y·Dc,yαCηx,y+∂55Fu∗x,y·Dc,yαCηx,y2dxdy≥0
Assume that the Legendre condition is violated at some(x0,y0)∈D; i.e., (29)∂55Fu∗x,y<0 Then, there exists a rectangle [a¯,b¯]×[c¯,d¯]≜D¯∈D and six real constants C1,C2,C3,C4,C5,C6 with C6<0, such that (30)∂33Fu∗x,y≤C1,∂34Fu∗x,y≤C2,∂35Fu∗x,y≤C3,∂44Fu∗x,y≤C4,∂45Fu∗x,y≤C5,∂55Fu∗x,y≤C6<0, for all (x,y)∈D¯.
Define the functionh(x,y):[a¯,b¯]×[c¯,d¯]→R as follows: (31)hs,t=α+2s-a¯1+α-2α+4b¯-a¯s-a¯2+α+α+10b¯-a¯2s-a¯3+α-4b¯-a¯3s-a¯4+α·α+2t-c¯1+α-2α+4d¯-c¯t-c¯2+α+α+10d¯-c¯2t-c¯3+α-4d¯-c¯3t-c¯4+α Then, h(s,t) is of class C1, and (32)ha¯,t=0=hb¯,t,t∈c¯,d¯hs,c¯=0=hs,d¯,s∈a¯,b¯hs′a¯,t=0=hs′b¯,t,t∈c¯,d¯ht′s,c¯=0=ht′s,d¯,s∈a¯,b¯Da¯,xαCha¯,t=0=Da¯,xαChb¯,t,t∈c¯,d¯Dc¯,yαChs,c¯=0=Dc¯,yαChs,d¯,s∈a¯,b¯
Moreover, for every(x,y)∈D¯, (33)hx,y≤142b¯-a¯1+αd¯-c¯1+αDa¯,xαChx,y≤C¯b¯-a¯d¯-c¯1+α and (34)Dc¯,yαChx,y≤C¯b¯-a¯1+αd¯-c¯,C¯=700
Let(35)η¯x,y=hx,y,x,y∈D¯0,otherwise
By the properties of functionh(x,y), we have η¯(x,y)∈C1(D),η¯∂D=0, and (36)Da,xαCη¯x,y=Da¯,xαChx,y,x,y∈D¯0,otherwiseDc,yαCη¯x,y=Dc¯,yαChx,y,x,y∈D¯0,otherwise
Bringing this variation into (28), we get (37)0≤∬D∂33Fu∗x,y·η¯2x,y+2∂34Fu∗x,y·η¯x,y·Da,xαCη¯x,y+2∂35Fu∗x,y·η¯x,y·Dc,yαCη¯x,y+∂44Fu∗x,y·Da,xαCη¯x,y2+2∂45Fu∗x,y·Da,xαCη¯x,y·Dc,yαCη¯x,y+∂55Fu∗x,y·Dc,yαCη¯x,y2dxdy=∬D¯∂33Fu∗x,y·h2x,y+2∂34Fu∗x,y·hx,y·Da,xαChx,y+2∂35Fu∗x,y·hx,y·Dc,yαChx,y+∂44Fu∗x,y·Da,xαChx,y2+2∂45Fu∗x,y·Da,xαChx,y·Dc,yαChx,y+∂55Fu∗x,y·Dc,yαChx,y2dxdy≤∬D¯144C1b¯-a¯2+2αd¯-c¯2+2α+2142C2C¯b¯-a¯2+αd¯-c¯2+2α+2142C3C¯b¯-a¯2+2αd¯-c¯2+α+C4C¯2b¯-a¯2d¯-c¯2+2α+2C5C¯2b¯-a¯2+αd¯-c¯2+α+C6C¯2b¯-a¯2+2αd¯-c¯2dxdy≤d¯-c¯2∬D¯144C1b¯-a¯2+2αd¯-c¯2α+2142C2C¯b¯-a¯2+αd¯-c¯2α+2142C3C¯b¯-a¯2+2αd¯-c¯α+C4C¯2b¯-a¯2d¯-c¯2α+2C5C¯2b¯-a¯2+αd¯-c¯α+C6C¯2b¯-a¯2+2αdxdy<0 If d¯-c¯≪b¯-a¯≪1, there arises a contradiction.
Hence, (27) holds. In the same manner, we can get (26).


### 3.3. The Fractional Variational Problem with Holonomic Constraint

Consider the functionalJ defined by (38)Ju1x,y,u2x,y=∬DFx,y,u1x,y,u2x,y,Da,xαCu1x,y,Da,xαCu2x,y,Dc,yαCu1x,y,Dc,yαCu2x,ydxdyon the space (39)U~=u1x,y,u2x,y∈C1D×C1D subject to (40)u1x,c=φ1x,x∈a,b;u1a,y=ψ1y,y∈c,d;φ1a=ψ1c;u2x,c=φ2x,x∈a,b;u2a,y=ψ2y,y∈c,d;φ2a=ψ2c, where φi∈AC([a,b]),(i=1,2) and ψi∈AC([c,d]),(i=1,2). For simplicity, we denote (41)ux,y=u1x,y,u2x,yDa,xαCux,y=Da,xαCu1x,y,Da,xαCu2x,yDx,bαCux,y=Dx,bαCu1x,y,Dx,bαCu2x,yDc,yαCux,y=Dc,yαCu1x,y,Dc,yαCu2x,yDy,dαCux,y=Dy,dαCu1x,y,Dy,dαCu2x,yux,y=x,y,u1x,y,u2x,y,Da,xαCu1x,y,Da,xαCu2x,y,Dc,yαCu1x,y,Dc,yαCu2x,yAssume the Lagrangian in (38) satisfies the following conditions:(iii)F:D×R6→R is continuously differentiable with respect to its ith argument, for i=3,4,⋯8;(iv) Given any functionsu1(x,y),u2(x,y)∈D, the maps (42)x,y↦Dx,bαC∂4Fx,y,ux,y,Da,xαCux,y,Dc,yαCux,yx,y↦Dy,dαC∂5Fx,y,ux,y,Da,xαCux,y,Dc,yαCux,y are continuous.We consider the variational problem when in presence of a holonomic constraint. Assume that the admissible functions lie on the surface(43)gx,y,u1x,y,u2x,y=0 where g:D×R2→R is continuously differentiable with respect to its ith argument, for i=3,4.Theorem 15.
Letu∗∈U be a minimizer of J as in (38), under the constraint (43). If (44)∂4gx,y,u1∗x,y,u2∗x,y≠0,∀x,y∈D, then there is a continuous function λ:D→R such that (45)∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,y+λx,y∂3gx,y,u1∗x,y,u2∗x,y=0and (46)∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,y+λx,y∂4gx,y,u1∗x,y,u2∗x,y=0Proof.
Consider a variation ofu∗(x,y) of type u∗+εη, with ε≪1, and η=(η1,η2)∈C1(D)×C1(D), satisfying the boundary conditions ηi(x,y)∂D=0,(i=1,2). By (47)∂4gx,y,u1∗x,y,u2∗x,y≠0,∀x,y∈D and the implicit function theorem, there exists a subfamily of variations satisfying restriction equation (43). That is, there exists a unique function η2(ε,η1) such that (u1∗+εη1,u2∗+εη2) satisfies (43). Hence, for all (x,y)∈D, we have (48)gx,y,u1∗x,y+εη1x,y,u2∗x,y+εη2x,y=0.
Differentiating (48) with respect to ε and putting ε=0, we get (49)∂3gx,y,u1∗x,y,u2∗x,yη1x,y+∂4gx,y,u1∗x,y,u2∗x,yη2x,y=0. Define the function (50)λx,y=-∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,y∂4gx,y,u1∗x,y,u2∗x,y
On the other hand, sinceu∗ is a minimizer of J, the first variation of J must vanish; (51)∬D∂3Fu∗x,y·η1x,y+∂4Fu∗x,y·η2x,y+∂5Fu∗x,y·Da,xαCη1x,y+∂6Fu∗x,y·Da,xαCη2x,y+∂7Fu∗x,y·Dc,yαCη1x,y+∂8Fu∗x,y·Dc,yαCη2x,ydxdy=0
Integrating by parts andηi(x,y)∂D=0,(i=1,2), we obtain (52)∬D∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,yη1x,y+∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,yη2x,ydxdy+∬D∂∂xη1x,y·Ix,b1-α∂5F[u∗x,y]+∂∂yη1x,y·Iy,d1-α∂7F[u∗x,y]dxdy+∬D∂∂xη2x,y·Ix,b1-α∂6F[u∗x,y]+∂∂yη2x,y·Iy,d1-α∂8F[u∗x,y]dxdy=∬D∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,yη1x,y+∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,yη2x,ydxdy+∮∂D-η1x,yIy,d1-α∂7Fu∗x,ydx+η1x,yIx,b1-α∂5Fu∗x,ydy+∮∂D-η2x,yIy,d1-α∂8Fu∗x,ydx+η2x,yIx,b1-α∂6Fu∗x,ydy=∬D∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,yη1x,y+∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,yη2x,ydxdy=∬D∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,y+λx,y∂3gx,y,u∗x,yη1x,ydxdy=0
Sinceη1(x,y) is arbitrary, we have that u∗ is a solution of the equation (53)∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,y+λx,y∂3gx,y,u∗x,y=0
At the same time, by (50), we obtain the second condition (54)∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,y+λx,y∂4gx,y,u∗x,y=0


### 3.4. Isoperimetric Problem

Letc be constant and N:D×R3→R be a continuously differentiable function with respect to its i-th argument, for i=3,4,5. For any function u∈C1(D), the maps (55)x,y↦Dx,bαC∂4Nx,y,ux,y,Da,xαCux,y,Dc,yαCux,y and (56)x,y↦Dy,dαC∂5Nx,y,ux,y,Da,xαCux,y,Dc,yαCux,yare continuous.We now consider a new isoperimetric type problem with the isoperimetric constraint as follows:(57)nux,y=∬DNx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy=cTheorem 16.
Suppose thatu∗ is a local minimizer for J as in (25) on the space D, subject to the integral constraint (57). If u∗ is not an extremal of n(u), then there exists a real λ such that u∗ is a solution of the equation (58)∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,y=0 where the function g:D×R3→R is defined by g=F+λN.Proof.
Consider a variation of two parametersu∗+ε1η1(x,y)+ε2η2(x,y), with εi≪1 and ηi(x,y)∈C1(D) satisfying ηi(x,y)∂D=0,i=1,2. Define the functions i and j with two parameters (ε1,ε2) in a neighborhood of zero as (59)iε1,ε2=nu∗+ε1η1x,y+ε2η2x,y-c and (60)jε1,ε2=Ju∗+ε1η1x,y+ε2η2x,y
Whenε1=ε2=0, the derivative ∂i/∂ε2 is (61)∂i∂ε20,0=∬D∂3Nu∗x,y·η2x,y+∂4Nu∗x,y·Da,xαCη2x,y+∂5Nu∗x,y·Dc,yαCη2x,ydxdy
Integrating by parts andηi(x,y)∂D=0,(i=1,2), we obtain(62)∂i∂ε20,0=∬D∂3Nu∗x,y-Dx,bα∂4Nu∗x,y-Dy,dα∂5Nu∗x,yη2x,ydxdy+∬D∂∂xIx,b1-α∂4Nu∗x,y·η2x,y+∂∂yIy,d1-α∂5Nu∗x,y·η2x,ydxdy=∬D∂3Nu∗x,y-Dx,bα∂4Nu∗x,y-Dy,dα∂5Nu∗x,yη2x,ydxdy+∮∂D-η2x,yIy,d1-α∂5Nu∗x,ydx+η2x,yIx,b1-α∂4Nu∗x,ydy=∬D∂3Nu∗x,y-Dx,bα∂4Nu∗x,y-Dy,dα∂5Nu∗x,yη2x,ydxdy
Sinceu∗ is not an extremal of n(u), there exists a function η2(x,y) such that ∂i/∂ε2(0,0)≠0. An application of the implicit function theorem implies that there exists a subfamily of variations satisfying the integral constraint.
On the other hand,(0,0) is a minimize of j, under the restriction i(·,·)=0, and we just proved that ∇i(0,0)≠0. By the Lagrange multiplier rule, there exists a real number λ such that ∇(j+λi)(0,0)=(0,0). In particular, ∂/∂ε1(j+λi)(0,0)=0. Since η1(x,y)∂D=0, we have (63)∬D∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,yη1x,ydxdy+∮∂D-η1x,yIy,d1-α∂5Nu∗x,ydx+η1x,yIx,b1-α∂4Nu∗x,ydy=∬D∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,yη1x,ydxdy=0
Hence,(64)∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,y=0
This ends the proof.Theorem 17.
Suppose that the functionsF(x,y,u(x,y),Da,xαCu(x,y),Dc,yαCu(x,y)) and N(x,y,u(x,y),Da,xαCu(x,y),Dc,yαCu(x,y)) as in (25) and (57) are convex in D×R3. Let λ≥0 be a constant and g=F+λN. Then, each solution u∗ of the fractional Euler-Lagrange equation (65)∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,y=0 minimizes J in U, subject to the integral constraint (57).Proof.
Consider a variation ofu∗(x,y) of type u∗+εη, with ε≪1 and η∈C1(D), satisfying the boundary conditions η(x,y)∂D=0. Noting that g is convex, (66)∬Dgx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy-∬Dgx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy≥∬D∂3gu∗x,yεηx,y+∂4gu∗x,yεDa,xαCηx,y+∂5gu∗x,yεDc,yαCηx,ydxdy=∬D∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,yεηx,ydxdy+∬D∂∂xIx,b1-α∂4gu∗x,yεηx,y+∂∂yIy,d1-α∂5gu∗x,yεηx,ydxdy=∬D∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,yεηx,ydxdy+ε∮∂D-Iy,d1-α∂5gu∗x,yηx,ydx+Ix,b1-α∂4gu∗x,yηx,ydy=0
We have(67)∬DFx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy+∬DλNx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy≥∬DFx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy+∬DλNx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy
According to the integral constraint, we obtain(68)∬DFx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy+λc≥∬DFx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy+λc
As a consequence,(69)∬DFx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy≥∬DFx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy
This ends the proof.


### 3.5. Noether Symmetry

Suppose that the configuration of a dynamical system is(70)Fx,y,ux,y,Da,xαCux,y,Dc,yαCux,yThe action is defined by (25); i.e., (71)J=∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdyThe isochronal variation principle(72)δJ=0with fixed end-point conditions (73)δu∂D=0is called the Hamilton principle on the action. From the principle equations (72)-(73) and Theorem 9, it is easy to obtain Euler-Lagrange equations 20′. Let us introduce the infinitesimal transformations of the one-parameter finite transformation group G; their expansion formulas are (74)x¯=x+εξxx,y,u,y¯=y+εξyx,y,u,u¯x¯,y¯=ux,y+εξux,y,uwhere ε is infinitesimal parameter; ξx,ξy, and ξu are infinitesimal generators or generating functions of the infinitesimal transformations. We introduce the following notation [14], Δx=d/dεε=0(x¯-x),Δy=d/dεε=0(y¯-y), and Δu=d/dεε=0(u¯(x¯,y¯)-u(x,y)).So we have(75)Δu=ddεε=0u¯x¯,y¯-ux,y=ddεε=0u¯x¯,y¯-u¯x,y¯+ddεε=0u¯x,y¯-u¯x,y+ddεε=0u¯x,y-ux,y=uxΔx+uyΔy+δu and (76)δu=Δu-uxΔx-uyΔy=ξu-uxξx-uyξyTheorem 18.
Letu∈AC(D) and G be a local one-parameter group of transformations by (74). Then (77)ΔDa,xαCux,y=Da,xαCδu+Da,xαuxx,y·ξxx,y,u+ddyDa,xαCux,y·ξyx,y,u+1Γ1-αuxa,yx-aα·ξxx,y,uand (78)ΔDc,yαCux,y=Dc,yαCδu+Dc,yαuyx,y·ξyx,y,u+ddxDc,yαCux,y·ξxx,y,u+1Γ1-αuyx,cy-cα·ξyx,y,uProof.
By the definitions, we have(79)ΔDa,xαCux,y=ddεσεσ=0Da¯,x¯αCu¯x¯,y¯-Da,xαCux,y=1Γ1-αddεε=0∫a¯x¯x¯-s-αu¯ss,y¯ds-∫axx-s-αuss,yds=1Γ1-αddεε=0∫a¯ax¯-s-αu¯ss,y¯ds+∫ax¯x¯-s-αu¯ss,y¯ds-∫axx-s-αu¯ss,y¯ds+∫axx-s-αu¯ss,y¯ds-∫axx-s-αu¯ss,yds+∫axx-s-αu¯ss,yds-∫axx-s-αuss,yds=Da,xαCδu+1Γ1-αddεε=0∫a¯ax¯-s-αu¯ss,y¯ds+1Γ1-αddεε=0∫ax¯x¯-s-αu¯ss,y¯ds-∫axx-s-αu¯ss,y¯ds+1Γ1-αddεε=0∫axx-s-αu¯ss,y¯ds-∫axx-s-αu¯ss,yds=Da,xαCδu+1Γ1-αddεε=0∫a¯ax¯-s-αu¯ss,y¯ds+ddx¯dx¯dεε=0Ia,x¯αu¯xx¯,y¯-Ia,xαu¯xx,y¯+ddy¯dy¯dεε=0Da,xαCu¯x,y¯-Da,xαCu¯x,y=Da,xαCδu+Da,xαuxx,y·ξxx,y,u+ddyDa,xαCux,y·ξyx,y,u+1Γ1-αuxa,yx-aα·ξxx,y,u
Analogously, we obtain(80)ΔDc,yαCux,y=Dc,yαCδu+Dc,yαuyx,y·ξyx,y,u+ddxDc,yαCux,y·ξxx,y,u+1Γ1-αuyx,cy-cα·ξyx,y,uUnder the infinitesimal transformations, the action equation (25) will be transformed into (81)JS¯=∬D¯Fx¯,y¯,u¯x¯,y¯,Da¯,x¯αCu¯x¯,y¯,Dc¯,y¯αCu¯x¯,y¯dx¯dy¯The difference of the fractional action before and after the infinitesimal transformation is(82)JS¯-JS=∬D¯Fx¯,y¯,u¯x¯,y¯,Da¯,x¯αCu¯x¯,y¯,Dc¯,y¯αCu¯x¯,y¯dx¯dy¯-∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy where the space S¯ is adjacent to the space S.Moreover,(83)ΔJ=ddεε=0∬D¯Fx¯,y¯,u¯x¯,y¯,Da¯,x¯αCu¯x¯,y¯,Dc¯,y¯αCu¯x¯,y¯dx¯dy¯-∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy=ddεε=0∬DFx¯,y¯,u¯x¯,y¯,Da¯,x¯αCu¯x¯,y¯,Dc¯,y¯αCu¯x¯,y¯det∂x¯,y¯∂x,y-Fx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy=∬D∂1FΔx+∂2FΔy+∂3FΔu+∂4FΔDa,xαCux,y+∂5FΔDc,yαCux,y+F·divX¯dxdy where (84)divX¯=ddεσdet∂x¯,y¯∂x,yεσ=0By Theorem9 and relation (76), we get (85)ΔJ=∬D∂1Fξx+∂2Fξy+∂3Fξu+∂4F·Da,xαCδu+Da,xαux·ξx+ddyDa,xαCu·ξy+1Γ1-αuxa,yx-aα·ξx+∂5F·Dc,yαCδu+Dc,yαuy·ξy+ddxDc,yαCu·ξx+1Γ1-αuyx,cy-cα·ξy+F·divX¯dxdy=∬D∂1F+ddxDc,yαCu·∂5F+∂4F·Da,xαux+1Γ1-αuxa,yx-aα·∂4Fξx+∂2F+ddyDa,xαCu·∂4F+∂5F·Dc,yαuy+1Γ1-αuyx,cy-cα·∂5Fξy+∂3F·ξu-Dx,bα∂4F·δu-Dy,dα∂5F·δu+F·divX¯dxdy+∬D∂∂xδu·Ix,b1-α∂4F+∂∂yδu·Iy,d1-α∂5Fdxdy=∬D∂1F+ddxDc,yαCu·∂5F+∂4F·Da,xαux+1Γ1-αuxa,yx-aα·∂4Fξx+∂2F+ddyDa,xαCu·∂4F+∂5F·Dc,yαuy+1Γ1-αuyx,cy-cα·∂5Fξy+∂3F·ξu-Dx,bα∂4F·δu-Dy,dα∂5F·δu+F·divX¯dxdy+∮∂D-δu·Iy,d1-α∂5Fdx+δu·Ix,b1-α∂4Fdy=∬D∂1F+ddxDc,yαCu·∂5F+∂4F·Da,xαux+ux·Dx,bα∂4F+ux·Dy,dα∂5F+1Γ1-αuxa,yx-aα·∂4Fξx+∂2F+ddyDa,xαCu·∂4F+∂5F·Dc,yαuy+uy·Dx,bα∂4F+uy·Dy,dα∂5F+1Γ1-αuyx,cy-cα·∂5Fξy+∂3F-Dx,bα∂4F-Dy,dα∂5Fξu+F·divX¯dxdy+∮∂D-δu·Iy,d1-α∂5Fdx+δu·Ix,b1-α∂4FdyDefinition 19.
If the fractional action is invariant under the infinitesimal transformations (74) of group, i.e., for each of the infinitesimal transformations, the formula (86)ΔJ=0 holds, then the infinitesimal transformations are called the symmetric transformations in sense of Noether for the fractional system.
By Definition19 and (73) and (85), we can get the following theorem.Theorem 20.
For the infinitesimal transformations (74) of group, if the condition (87)∂1F+ddxDc,yαCu·∂5F+∂4F·Da,xαux+ux·Dx,bα∂4F+ux·Dy,dα∂5F+1Γ1-αuxa,yx-aα·∂4Fξx+∂2F+ddyDa,xαCu·∂4F+∂5F·Dc,yαuy+uy·Dx,bα∂4F+uy·Dy,dα∂5F+1Γ1-αuyx,cy-cα·∂5Fξy+∂3F-Dx,bα∂4F-Dy,dα∂5Fξu+F·divX¯=0is satisfied, then the transformations (74) are the Noether symmetric transformations for the dynamical system based on the action.


## 3.1. The Fundamental Problem

Theorem 11.
Suppose thatu∗(x,y) is a local minimizer for J as in (17), defined on (19)U=ux,y∈C1D∣ux,c=φx,ua,y=ψy,x∈a,b,y∈c,d where φ(x)∈AC([a,b]) and ψ(y)∈AC([c,d]). Then u∗(x,y) is a solution to the equation (20)∂3Fu∗x,y-∂∂x∂4Fu∗x,y-∂∂y∂5Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂7Fu∗x,y=0for almost all (x,y)∈D.Proof.
Letu(x,y)+εη(x,y) be a variation of u(x,y), with ε≪1, ηx,y∈C1D∩U, and η(x,y)∂D=0. We define the function h in a neighborhood of zero by the expression (21)hε=Ju∗+εη
Sinceu∗ is a minimizer of J, then ε=0 is a minimizer of h, and so h′(0)=0. Differentiating h at ε=0 and using Theorem 9, we obtain(22)h′0=∬D∂3Fu∗x,y·ηx,y+∂4Fu∗x,y·∂xηx,y+∂5Fu∗x,y·∂yηx,y+∂6Fu∗x,y·Da,xαCηx,y+∂7Fu∗x,y·Dc,yαCηx,ydxdy⇒h′0=∬D∂3Fu∗x,y-∂∂x∂4Fu∗x,y-∂∂y∂5Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂7Fu∗x,yηx,ydxdy+∬D∂∂x∂4Fu∗x,y+Ix,b1-α∂6Fu∗x,yηx,ydxdy+∬D∂∂y∂5Fu∗x,y+Iy,d1-α∂7Fu∗x,yηx,ydxdy⇒h′0=∬D∂3Fu∗x,y-∂∂x∂4Fu∗x,y-∂∂y∂5Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂7Fu∗x,yηx,ydxdy-∮∂D∂5Fu∗x,y+Iy,d1-α∂7Fu∗x,yηx,ydx+∮∂D∂4Fu∗x,y+Ix,b1-α∂6Fu∗x,yηx,ydy⇒h′0=0.
Sinceη(x,y)∂D=0 and η(x,y) is arbitrary elsewhere, we conclude that (23)∂3Fu∗x,y-∂∂x∂4Fu∗x,y-∂∂y∂5Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂7Fu∗x,y=0 for almost all (x,y)∈D.Remark 12.
Equation (20) is called the Euler-Lagrange equation associated with the functional J. Solutions of this equation are called extremals.Remark 13.
If we define functional(24)Jux,y=∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy then we obtain the Euler-Lagrange equation 20′∂3Fu∗x,y-Dx,bα∂4Fu∗x,y-Dy,dα∂5Fu∗x,y=0for almost all (x,y)∈D.


## 3.2. The Legendre Condition

In [17], a second-order necessary condition had been proved for functionals involving Riemann-Liouville fractional derivatives.In this section, we give a second-order necessary condition, usually called Legendre condition, for functionals involving Riemann-Liouville partial fractional derivatives.We introduce the functional(25)Jux,y=∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdywith the same assumptions on F as in Section 3.1.Theorem 14.
Suppose thatu∗ is a local minimizer for J in U. If ∂ij2F exists and is continuous for i,j∈{3,4,5}, then u∗ satisfies (26)∂44Fu∗x,y≥0(27)∂55Fu∗x,y≥0for almost all (x,y)∈D.Proof.
Letu∗(x,y)+εη(x,y) be a variation of u∗(x,y), with ε≪1, ηx,y∈C1D∩U, and η(x,y)∂D=0.
We define the functionh in a neighborhood of zero by the expression h(ε)=J(u∗+εη). Then, we have h′′(0)≥0; that is, (28)∬D∂33Fu∗x,y·η2x,y+2∂34Fu∗x,y·ηx,y·Da,xαCηx,y+2∂35Fu∗x,y·ηx,y·Dc,yαCηx,y+∂44Fu∗x,y·Da,xαCηx,y2+2∂45Fu∗x,y·Da,xαCηx,y·Dc,yαCηx,y+∂55Fu∗x,y·Dc,yαCηx,y2dxdy≥0
Assume that the Legendre condition is violated at some(x0,y0)∈D; i.e., (29)∂55Fu∗x,y<0 Then, there exists a rectangle [a¯,b¯]×[c¯,d¯]≜D¯∈D and six real constants C1,C2,C3,C4,C5,C6 with C6<0, such that (30)∂33Fu∗x,y≤C1,∂34Fu∗x,y≤C2,∂35Fu∗x,y≤C3,∂44Fu∗x,y≤C4,∂45Fu∗x,y≤C5,∂55Fu∗x,y≤C6<0, for all (x,y)∈D¯.
Define the functionh(x,y):[a¯,b¯]×[c¯,d¯]→R as follows: (31)hs,t=α+2s-a¯1+α-2α+4b¯-a¯s-a¯2+α+α+10b¯-a¯2s-a¯3+α-4b¯-a¯3s-a¯4+α·α+2t-c¯1+α-2α+4d¯-c¯t-c¯2+α+α+10d¯-c¯2t-c¯3+α-4d¯-c¯3t-c¯4+α Then, h(s,t) is of class C1, and (32)ha¯,t=0=hb¯,t,t∈c¯,d¯hs,c¯=0=hs,d¯,s∈a¯,b¯hs′a¯,t=0=hs′b¯,t,t∈c¯,d¯ht′s,c¯=0=ht′s,d¯,s∈a¯,b¯Da¯,xαCha¯,t=0=Da¯,xαChb¯,t,t∈c¯,d¯Dc¯,yαChs,c¯=0=Dc¯,yαChs,d¯,s∈a¯,b¯
Moreover, for every(x,y)∈D¯, (33)hx,y≤142b¯-a¯1+αd¯-c¯1+αDa¯,xαChx,y≤C¯b¯-a¯d¯-c¯1+α and (34)Dc¯,yαChx,y≤C¯b¯-a¯1+αd¯-c¯,C¯=700
Let(35)η¯x,y=hx,y,x,y∈D¯0,otherwise
By the properties of functionh(x,y), we have η¯(x,y)∈C1(D),η¯∂D=0, and (36)Da,xαCη¯x,y=Da¯,xαChx,y,x,y∈D¯0,otherwiseDc,yαCη¯x,y=Dc¯,yαChx,y,x,y∈D¯0,otherwise
Bringing this variation into (28), we get (37)0≤∬D∂33Fu∗x,y·η¯2x,y+2∂34Fu∗x,y·η¯x,y·Da,xαCη¯x,y+2∂35Fu∗x,y·η¯x,y·Dc,yαCη¯x,y+∂44Fu∗x,y·Da,xαCη¯x,y2+2∂45Fu∗x,y·Da,xαCη¯x,y·Dc,yαCη¯x,y+∂55Fu∗x,y·Dc,yαCη¯x,y2dxdy=∬D¯∂33Fu∗x,y·h2x,y+2∂34Fu∗x,y·hx,y·Da,xαChx,y+2∂35Fu∗x,y·hx,y·Dc,yαChx,y+∂44Fu∗x,y·Da,xαChx,y2+2∂45Fu∗x,y·Da,xαChx,y·Dc,yαChx,y+∂55Fu∗x,y·Dc,yαChx,y2dxdy≤∬D¯144C1b¯-a¯2+2αd¯-c¯2+2α+2142C2C¯b¯-a¯2+αd¯-c¯2+2α+2142C3C¯b¯-a¯2+2αd¯-c¯2+α+C4C¯2b¯-a¯2d¯-c¯2+2α+2C5C¯2b¯-a¯2+αd¯-c¯2+α+C6C¯2b¯-a¯2+2αd¯-c¯2dxdy≤d¯-c¯2∬D¯144C1b¯-a¯2+2αd¯-c¯2α+2142C2C¯b¯-a¯2+αd¯-c¯2α+2142C3C¯b¯-a¯2+2αd¯-c¯α+C4C¯2b¯-a¯2d¯-c¯2α+2C5C¯2b¯-a¯2+αd¯-c¯α+C6C¯2b¯-a¯2+2αdxdy<0 If d¯-c¯≪b¯-a¯≪1, there arises a contradiction.
Hence, (27) holds. In the same manner, we can get (26).


## 3.3. The Fractional Variational Problem with Holonomic Constraint

Consider the functionalJ defined by (38)Ju1x,y,u2x,y=∬DFx,y,u1x,y,u2x,y,Da,xαCu1x,y,Da,xαCu2x,y,Dc,yαCu1x,y,Dc,yαCu2x,ydxdyon the space (39)U~=u1x,y,u2x,y∈C1D×C1D subject to (40)u1x,c=φ1x,x∈a,b;u1a,y=ψ1y,y∈c,d;φ1a=ψ1c;u2x,c=φ2x,x∈a,b;u2a,y=ψ2y,y∈c,d;φ2a=ψ2c, where φi∈AC([a,b]),(i=1,2) and ψi∈AC([c,d]),(i=1,2). For simplicity, we denote (41)ux,y=u1x,y,u2x,yDa,xαCux,y=Da,xαCu1x,y,Da,xαCu2x,yDx,bαCux,y=Dx,bαCu1x,y,Dx,bαCu2x,yDc,yαCux,y=Dc,yαCu1x,y,Dc,yαCu2x,yDy,dαCux,y=Dy,dαCu1x,y,Dy,dαCu2x,yux,y=x,y,u1x,y,u2x,y,Da,xαCu1x,y,Da,xαCu2x,y,Dc,yαCu1x,y,Dc,yαCu2x,yAssume the Lagrangian in (38) satisfies the following conditions:(iii)F:D×R6→R is continuously differentiable with respect to its ith argument, for i=3,4,⋯8;(iv) Given any functionsu1(x,y),u2(x,y)∈D, the maps (42)x,y↦Dx,bαC∂4Fx,y,ux,y,Da,xαCux,y,Dc,yαCux,yx,y↦Dy,dαC∂5Fx,y,ux,y,Da,xαCux,y,Dc,yαCux,y are continuous.We consider the variational problem when in presence of a holonomic constraint. Assume that the admissible functions lie on the surface(43)gx,y,u1x,y,u2x,y=0 where g:D×R2→R is continuously differentiable with respect to its ith argument, for i=3,4.Theorem 15.
Letu∗∈U be a minimizer of J as in (38), under the constraint (43). If (44)∂4gx,y,u1∗x,y,u2∗x,y≠0,∀x,y∈D, then there is a continuous function λ:D→R such that (45)∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,y+λx,y∂3gx,y,u1∗x,y,u2∗x,y=0and (46)∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,y+λx,y∂4gx,y,u1∗x,y,u2∗x,y=0Proof.
Consider a variation ofu∗(x,y) of type u∗+εη, with ε≪1, and η=(η1,η2)∈C1(D)×C1(D), satisfying the boundary conditions ηi(x,y)∂D=0,(i=1,2). By (47)∂4gx,y,u1∗x,y,u2∗x,y≠0,∀x,y∈D and the implicit function theorem, there exists a subfamily of variations satisfying restriction equation (43). That is, there exists a unique function η2(ε,η1) such that (u1∗+εη1,u2∗+εη2) satisfies (43). Hence, for all (x,y)∈D, we have (48)gx,y,u1∗x,y+εη1x,y,u2∗x,y+εη2x,y=0.
Differentiating (48) with respect to ε and putting ε=0, we get (49)∂3gx,y,u1∗x,y,u2∗x,yη1x,y+∂4gx,y,u1∗x,y,u2∗x,yη2x,y=0. Define the function (50)λx,y=-∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,y∂4gx,y,u1∗x,y,u2∗x,y
On the other hand, sinceu∗ is a minimizer of J, the first variation of J must vanish; (51)∬D∂3Fu∗x,y·η1x,y+∂4Fu∗x,y·η2x,y+∂5Fu∗x,y·Da,xαCη1x,y+∂6Fu∗x,y·Da,xαCη2x,y+∂7Fu∗x,y·Dc,yαCη1x,y+∂8Fu∗x,y·Dc,yαCη2x,ydxdy=0
Integrating by parts andηi(x,y)∂D=0,(i=1,2), we obtain (52)∬D∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,yη1x,y+∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,yη2x,ydxdy+∬D∂∂xη1x,y·Ix,b1-α∂5F[u∗x,y]+∂∂yη1x,y·Iy,d1-α∂7F[u∗x,y]dxdy+∬D∂∂xη2x,y·Ix,b1-α∂6F[u∗x,y]+∂∂yη2x,y·Iy,d1-α∂8F[u∗x,y]dxdy=∬D∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,yη1x,y+∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,yη2x,ydxdy+∮∂D-η1x,yIy,d1-α∂7Fu∗x,ydx+η1x,yIx,b1-α∂5Fu∗x,ydy+∮∂D-η2x,yIy,d1-α∂8Fu∗x,ydx+η2x,yIx,b1-α∂6Fu∗x,ydy=∬D∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,yη1x,y+∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,yη2x,ydxdy=∬D∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,y+λx,y∂3gx,y,u∗x,yη1x,ydxdy=0
Sinceη1(x,y) is arbitrary, we have that u∗ is a solution of the equation (53)∂3Fu∗x,y-Dx,bα∂5Fu∗x,y-Dy,dα∂7Fu∗x,y+λx,y∂3gx,y,u∗x,y=0
At the same time, by (50), we obtain the second condition (54)∂4Fu∗x,y-Dx,bα∂6Fu∗x,y-Dy,dα∂8Fu∗x,y+λx,y∂4gx,y,u∗x,y=0


## 3.4. Isoperimetric Problem

Letc be constant and N:D×R3→R be a continuously differentiable function with respect to its i-th argument, for i=3,4,5. For any function u∈C1(D), the maps (55)x,y↦Dx,bαC∂4Nx,y,ux,y,Da,xαCux,y,Dc,yαCux,y and (56)x,y↦Dy,dαC∂5Nx,y,ux,y,Da,xαCux,y,Dc,yαCux,yare continuous.We now consider a new isoperimetric type problem with the isoperimetric constraint as follows:(57)nux,y=∬DNx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy=cTheorem 16.
Suppose thatu∗ is a local minimizer for J as in (25) on the space D, subject to the integral constraint (57). If u∗ is not an extremal of n(u), then there exists a real λ such that u∗ is a solution of the equation (58)∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,y=0 where the function g:D×R3→R is defined by g=F+λN.Proof.
Consider a variation of two parametersu∗+ε1η1(x,y)+ε2η2(x,y), with εi≪1 and ηi(x,y)∈C1(D) satisfying ηi(x,y)∂D=0,i=1,2. Define the functions i and j with two parameters (ε1,ε2) in a neighborhood of zero as (59)iε1,ε2=nu∗+ε1η1x,y+ε2η2x,y-c and (60)jε1,ε2=Ju∗+ε1η1x,y+ε2η2x,y
Whenε1=ε2=0, the derivative ∂i/∂ε2 is (61)∂i∂ε20,0=∬D∂3Nu∗x,y·η2x,y+∂4Nu∗x,y·Da,xαCη2x,y+∂5Nu∗x,y·Dc,yαCη2x,ydxdy
Integrating by parts andηi(x,y)∂D=0,(i=1,2), we obtain(62)∂i∂ε20,0=∬D∂3Nu∗x,y-Dx,bα∂4Nu∗x,y-Dy,dα∂5Nu∗x,yη2x,ydxdy+∬D∂∂xIx,b1-α∂4Nu∗x,y·η2x,y+∂∂yIy,d1-α∂5Nu∗x,y·η2x,ydxdy=∬D∂3Nu∗x,y-Dx,bα∂4Nu∗x,y-Dy,dα∂5Nu∗x,yη2x,ydxdy+∮∂D-η2x,yIy,d1-α∂5Nu∗x,ydx+η2x,yIx,b1-α∂4Nu∗x,ydy=∬D∂3Nu∗x,y-Dx,bα∂4Nu∗x,y-Dy,dα∂5Nu∗x,yη2x,ydxdy
Sinceu∗ is not an extremal of n(u), there exists a function η2(x,y) such that ∂i/∂ε2(0,0)≠0. An application of the implicit function theorem implies that there exists a subfamily of variations satisfying the integral constraint.
On the other hand,(0,0) is a minimize of j, under the restriction i(·,·)=0, and we just proved that ∇i(0,0)≠0. By the Lagrange multiplier rule, there exists a real number λ such that ∇(j+λi)(0,0)=(0,0). In particular, ∂/∂ε1(j+λi)(0,0)=0. Since η1(x,y)∂D=0, we have (63)∬D∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,yη1x,ydxdy+∮∂D-η1x,yIy,d1-α∂5Nu∗x,ydx+η1x,yIx,b1-α∂4Nu∗x,ydy=∬D∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,yη1x,ydxdy=0
Hence,(64)∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,y=0
This ends the proof.Theorem 17.
Suppose that the functionsF(x,y,u(x,y),Da,xαCu(x,y),Dc,yαCu(x,y)) and N(x,y,u(x,y),Da,xαCu(x,y),Dc,yαCu(x,y)) as in (25) and (57) are convex in D×R3. Let λ≥0 be a constant and g=F+λN. Then, each solution u∗ of the fractional Euler-Lagrange equation (65)∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,y=0 minimizes J in U, subject to the integral constraint (57).Proof.
Consider a variation ofu∗(x,y) of type u∗+εη, with ε≪1 and η∈C1(D), satisfying the boundary conditions η(x,y)∂D=0. Noting that g is convex, (66)∬Dgx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy-∬Dgx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy≥∬D∂3gu∗x,yεηx,y+∂4gu∗x,yεDa,xαCηx,y+∂5gu∗x,yεDc,yαCηx,ydxdy=∬D∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,yεηx,ydxdy+∬D∂∂xIx,b1-α∂4gu∗x,yεηx,y+∂∂yIy,d1-α∂5gu∗x,yεηx,ydxdy=∬D∂3gu∗x,y-Dx,bα∂4gu∗x,y-Dy,dα∂5gu∗x,yεηx,ydxdy+ε∮∂D-Iy,d1-α∂5gu∗x,yηx,ydx+Ix,b1-α∂4gu∗x,yηx,ydy=0
We have(67)∬DFx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy+∬DλNx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy≥∬DFx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy+∬DλNx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy
According to the integral constraint, we obtain(68)∬DFx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy+λc≥∬DFx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy+λc
As a consequence,(69)∬DFx,y,u∗x,y+εηx,y,Da,xαCu∗x,y+εDa,xαCηx,y,Dc,yαCu∗x,y+εDc,yαCηx,ydxdy≥∬DFx,y,u∗x,y,Da,xαCu∗x,y,Dc,yαCu∗x,ydxdy
This ends the proof.


## 3.5. Noether Symmetry

Suppose that the configuration of a dynamical system is(70)Fx,y,ux,y,Da,xαCux,y,Dc,yαCux,yThe action is defined by (25); i.e., (71)J=∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdyThe isochronal variation principle(72)δJ=0with fixed end-point conditions (73)δu∂D=0is called the Hamilton principle on the action. From the principle equations (72)-(73) and Theorem 9, it is easy to obtain Euler-Lagrange equations 20′. Let us introduce the infinitesimal transformations of the one-parameter finite transformation group G; their expansion formulas are (74)x¯=x+εξxx,y,u,y¯=y+εξyx,y,u,u¯x¯,y¯=ux,y+εξux,y,uwhere ε is infinitesimal parameter; ξx,ξy, and ξu are infinitesimal generators or generating functions of the infinitesimal transformations. We introduce the following notation [14], Δx=d/dεε=0(x¯-x),Δy=d/dεε=0(y¯-y), and Δu=d/dεε=0(u¯(x¯,y¯)-u(x,y)).So we have(75)Δu=ddεε=0u¯x¯,y¯-ux,y=ddεε=0u¯x¯,y¯-u¯x,y¯+ddεε=0u¯x,y¯-u¯x,y+ddεε=0u¯x,y-ux,y=uxΔx+uyΔy+δu and (76)δu=Δu-uxΔx-uyΔy=ξu-uxξx-uyξyTheorem 18.
Letu∈AC(D) and G be a local one-parameter group of transformations by (74). Then (77)ΔDa,xαCux,y=Da,xαCδu+Da,xαuxx,y·ξxx,y,u+ddyDa,xαCux,y·ξyx,y,u+1Γ1-αuxa,yx-aα·ξxx,y,uand (78)ΔDc,yαCux,y=Dc,yαCδu+Dc,yαuyx,y·ξyx,y,u+ddxDc,yαCux,y·ξxx,y,u+1Γ1-αuyx,cy-cα·ξyx,y,uProof.
By the definitions, we have(79)ΔDa,xαCux,y=ddεσεσ=0Da¯,x¯αCu¯x¯,y¯-Da,xαCux,y=1Γ1-αddεε=0∫a¯x¯x¯-s-αu¯ss,y¯ds-∫axx-s-αuss,yds=1Γ1-αddεε=0∫a¯ax¯-s-αu¯ss,y¯ds+∫ax¯x¯-s-αu¯ss,y¯ds-∫axx-s-αu¯ss,y¯ds+∫axx-s-αu¯ss,y¯ds-∫axx-s-αu¯ss,yds+∫axx-s-αu¯ss,yds-∫axx-s-αuss,yds=Da,xαCδu+1Γ1-αddεε=0∫a¯ax¯-s-αu¯ss,y¯ds+1Γ1-αddεε=0∫ax¯x¯-s-αu¯ss,y¯ds-∫axx-s-αu¯ss,y¯ds+1Γ1-αddεε=0∫axx-s-αu¯ss,y¯ds-∫axx-s-αu¯ss,yds=Da,xαCδu+1Γ1-αddεε=0∫a¯ax¯-s-αu¯ss,y¯ds+ddx¯dx¯dεε=0Ia,x¯αu¯xx¯,y¯-Ia,xαu¯xx,y¯+ddy¯dy¯dεε=0Da,xαCu¯x,y¯-Da,xαCu¯x,y=Da,xαCδu+Da,xαuxx,y·ξxx,y,u+ddyDa,xαCux,y·ξyx,y,u+1Γ1-αuxa,yx-aα·ξxx,y,u
Analogously, we obtain(80)ΔDc,yαCux,y=Dc,yαCδu+Dc,yαuyx,y·ξyx,y,u+ddxDc,yαCux,y·ξxx,y,u+1Γ1-αuyx,cy-cα·ξyx,y,uUnder the infinitesimal transformations, the action equation (25) will be transformed into (81)JS¯=∬D¯Fx¯,y¯,u¯x¯,y¯,Da¯,x¯αCu¯x¯,y¯,Dc¯,y¯αCu¯x¯,y¯dx¯dy¯The difference of the fractional action before and after the infinitesimal transformation is(82)JS¯-JS=∬D¯Fx¯,y¯,u¯x¯,y¯,Da¯,x¯αCu¯x¯,y¯,Dc¯,y¯αCu¯x¯,y¯dx¯dy¯-∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy where the space S¯ is adjacent to the space S.Moreover,(83)ΔJ=ddεε=0∬D¯Fx¯,y¯,u¯x¯,y¯,Da¯,x¯αCu¯x¯,y¯,Dc¯,y¯αCu¯x¯,y¯dx¯dy¯-∬DFx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy=ddεε=0∬DFx¯,y¯,u¯x¯,y¯,Da¯,x¯αCu¯x¯,y¯,Dc¯,y¯αCu¯x¯,y¯det∂x¯,y¯∂x,y-Fx,y,ux,y,Da,xαCux,y,Dc,yαCux,ydxdy=∬D∂1FΔx+∂2FΔy+∂3FΔu+∂4FΔDa,xαCux,y+∂5FΔDc,yαCux,y+F·divX¯dxdy where (84)divX¯=ddεσdet∂x¯,y¯∂x,yεσ=0By Theorem9 and relation (76), we get (85)ΔJ=∬D∂1Fξx+∂2Fξy+∂3Fξu+∂4F·Da,xαCδu+Da,xαux·ξx+ddyDa,xαCu·ξy+1Γ1-αuxa,yx-aα·ξx+∂5F·Dc,yαCδu+Dc,yαuy·ξy+ddxDc,yαCu·ξx+1Γ1-αuyx,cy-cα·ξy+F·divX¯dxdy=∬D∂1F+ddxDc,yαCu·∂5F+∂4F·Da,xαux+1Γ1-αuxa,yx-aα·∂4Fξx+∂2F+ddyDa,xαCu·∂4F+∂5F·Dc,yαuy+1Γ1-αuyx,cy-cα·∂5Fξy+∂3F·ξu-Dx,bα∂4F·δu-Dy,dα∂5F·δu+F·divX¯dxdy+∬D∂∂xδu·Ix,b1-α∂4F+∂∂yδu·Iy,d1-α∂5Fdxdy=∬D∂1F+ddxDc,yαCu·∂5F+∂4F·Da,xαux+1Γ1-αuxa,yx-aα·∂4Fξx+∂2F+ddyDa,xαCu·∂4F+∂5F·Dc,yαuy+1Γ1-αuyx,cy-cα·∂5Fξy+∂3F·ξu-Dx,bα∂4F·δu-Dy,dα∂5F·δu+F·divX¯dxdy+∮∂D-δu·Iy,d1-α∂5Fdx+δu·Ix,b1-α∂4Fdy=∬D∂1F+ddxDc,yαCu·∂5F+∂4F·Da,xαux+ux·Dx,bα∂4F+ux·Dy,dα∂5F+1Γ1-αuxa,yx-aα·∂4Fξx+∂2F+ddyDa,xαCu·∂4F+∂5F·Dc,yαuy+uy·Dx,bα∂4F+uy·Dy,dα∂5F+1Γ1-αuyx,cy-cα·∂5Fξy+∂3F-Dx,bα∂4F-Dy,dα∂5Fξu+F·divX¯dxdy+∮∂D-δu·Iy,d1-α∂5Fdx+δu·Ix,b1-α∂4FdyDefinition 19.
If the fractional action is invariant under the infinitesimal transformations (74) of group, i.e., for each of the infinitesimal transformations, the formula (86)ΔJ=0 holds, then the infinitesimal transformations are called the symmetric transformations in sense of Noether for the fractional system.
By Definition19 and (73) and (85), we can get the following theorem.Theorem 20.
For the infinitesimal transformations (74) of group, if the condition (87)∂1F+ddxDc,yαCu·∂5F+∂4F·Da,xαux+ux·Dx,bα∂4F+ux·Dy,dα∂5F+1Γ1-αuxa,yx-aα·∂4Fξx+∂2F+ddyDa,xαCu·∂4F+∂5F·Dc,yαuy+uy·Dx,bα∂4F+uy·Dy,dα∂5F+1Γ1-αuyx,cy-cα·∂5Fξy+∂3F-Dx,bα∂4F-Dy,dα∂5Fξu+F·divX¯=0is satisfied, then the transformations (74) are the Noether symmetric transformations for the dynamical system based on the action.


## 4. Examples

Example 21.
Consider the functional(88)Ju=∬DD0,xαCux,y-x1-αΓ2-α2+D0,yαCux,y-y1-αΓ2-α2+x2+y2dxdy subject to the restriction u(x,0)=0,u(0,y)=y and D=[0,b]×[0,d]. From Theorem 11, the necessary condition that every minimizer of the functional must fulfill is the following: (89)D0,xα2D0,xαCux,y-x1-αΓ2-α+D0,yα2D0,yαCux,y-y1-αΓ2-α=0
We see that(90)u∗=x+y satisfies these conditions. Also, using the Legendre condition, a local minimizer for the functional must verify (26) and (27), which for our example is verified. In fact, ∂44F[u∗]=2>0 and ∂44F[u∗]=2>0.Example 22.
Let(91)Ju=∬DD0,xαCux,y2+x1-αΓ2-α2+D0,yαCux,y-y1-αΓ2-α2+x2+y2dxdy subject to the restrictions u(x,0)=0,u(0,y)=y and (92)∬DD0,xαCux,y·x1-αΓ2-αdxdy=c where D=[0,b]×[0,d] and c=∬D‍x1-α/Γ(2-α)2dxdy=1/Γ(3-α).
The function(93)u∗=x+y satisfies the necessary conditions, as stated in Theorem 15.Example 23.
Let us consider the following fractional problem of the calculus of variations:(94)Ju=∬Dux,yDx,bα1+Dy,dα1+Da,xαCux,y+Dc,yαCux,ydxdy subject to the restriction u(x,c)=0,u(a,y)=y and D=[a,b]×[c,d].
If the transformation isξx,ξy,ξu=0,0,c, where c is arbitrary constant, then the fractional integral functional is invariant under Definition 19. In fact, the transformation satisfies (87).



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