IEEE/CAA Journal of Automatica Sinica
Citation: | Matheus J. Lazo and DelfimF.M. Torres, "Variational Calculus With Conformable Fractional Derivatives," IEEE/CAA J. Autom. Sinica, vol. 4, no. 2, pp. 340-352, Apr. 2017. doi: 10.1109/JAS.2016.7510160 |
FRACTIONAL calculus is a generalization of (integer) differential calculus, allowing to define integrals and derivatives of real or complex order [1]-[3]. It had its origin in the 1600 s and for three centuries the theory of fractional derivatives developed as a pure theoretical field of mathematics, useful only for mathematicians. The theory took more or less finished form by the end of the 19th century. In the last few decades, fractional differentiation has been "rediscovered" by applied scientists, proving to be very useful in various fields: physics (classic and quantum mechanics, thermodynamics, etc.), chemistry, biology, economics, engineering, signal and image processing, and control theory [4]. One can find in the existent literature several definitions of fractional derivatives, including the Riemann-Liouville, Caputo, Riesz, Riesz-Caputo, Weyl, Grunwald-Letnikov, Hadamard, and Chen derivatives. Recently, a simple solution to the discrepancies between known definitions was presented with the introduction of a new fractional notion, called the conformable derivative [5]. The new definition is a natural extension of the usual derivative, and satisfies the main properties one expects in a derivative: the conformable derivative of a constant is zero; satisfies the standard formulas of the derivative of the product and of the derivative of the quotient of two functions; and satisfies the chain rule. Besides simple and similar to the standard derivative, one can say that the conformable derivative combines the best characteristics of known fractional derivatives [6]. For this reason, the subject is now under strong development: see [7]-[10] and references therein.
The fractional calculus of variations was introduced in the context of classical mechanics when Riewe [11] showed that a Lagrangian involving fractional time derivatives leads to an equation of motion with non-conservative forces such as friction. It is a remarkable result since frictional and non-conservative forces are beyond the usual macroscopic variational treatment [12]. Riewe generalized the usual calculus of variations for a Lagrangian depending on Riemann-Liouville fractional derivatives [11] in order to deal with linear non-conservative forces. Actually, several approaches have been developed to generalize the calculus of variations to include problems depending on Caputo fractional derivatives, Riemann-Liouville fractional derivatives, Riesz fractional derivatives and others [13]-[19] (see [20]-[22] for the state of the art). Among theses approaches, recently it was shown that the action principle for dissipative systems can be generalized, fixing the mathematical inconsistencies present in the original Riewe's formulation, by using Lagrangians depending on classical and Caputo derivatives [23].
In this paper we work with conformable fractional derivatives in the context of the calculus of variations and optimal control [20]. In order to illustrate the potential application of conformable fractional derivatives in physical problems we show that it is possible to formulate an action principle with conformable fractional calculus for the frictional force free from the mathematical inconsistencies found in the Riewe original approach and far simpler than the formulations proposed in [23]. Furthermore, we obtain a generalization of Noether's symmetry theorem for fractional variational problems and we also consider conformable fractional optimal control problems. Emmy Noether was the first who proved, in 1918, that the notions of invariance and constant of motion are connected: when a system is invariant under a family of transformations, then a conserved quantity along the Euler-Lagrange extremals can be obtained [24], [25]. All conservation laws of Mechanics, e.g., conservation of energy or conservation of momentum, are easily explained from Noether's theorem. In this paper we study necessary conditions for invariance under a family of continuous transformations, where the Lagrangian contains a conformable fractional derivative of order α∈(0,1) . When α→1 , we obtain some well-known results, in particular the Noether theorem [25]. The advantages of our fractional results are clear. Indeed, the classical constants of motion appear naturally in closed systems while in practical terms closed systems do not exist: forces that do not store energy, so-called nonconservative or dissipative forces, are always present in real systems. Fractional dynamics provide a good way to model nonconservative systems [11]. Nonconservative forces remove energy from the systems and, as a consequence, the standard Noether constants of motion are broken [26]. Our results assert that it is still possible to obtain Noether-type theorems, which cover both conservative and nonconservative cases, and that this is done in a particularly simple and elegant way via the conformable fractional approach. This is in contrast with the approaches followed in [27]-[30].
The paper is organized as follows. In Section Ⅱ we collect some necessary definitions and results on the conformable fractional calculus needed in the sequel. In Section Ⅲ we obtain the conformable fractional Euler-Lagrange equation and in Section Ⅳ we formulate an action principle for dissipative systems, as an example of application and motivation to study the conformable calculus of variations. In Section V, we present an immediate consequence of the Euler-Lagrange equation that we use later in Sections Ⅵ and Ⅶ, where we prove, respectively, some necessary conditions for invariant fractional problems and a conformable fractional Noether theorem. We then review the obtained results using the Hamiltonian language in Section Ⅷ. In Section Ⅸ, we consider the conformable fractional optimal control problem, where the dynamic constraint is given by a conformable fractional derivative. Using the Hamiltonian language, we provide an invariant condition. In Section Ⅹ we consider the multi-dimensional case, for several independent and dependent variables.
In this section we review the conformable fractional calculus [5]-[7]. The conformable fractional derivative is a new well-behaved definition of fractional derivative, based on a simple limit definition. We review in this section the generalization of [5] proposed in [6].
Definition 1: The left conformable fractional derivative of order 0<α≤1 starting from a∈R of a function f:[a,b]→R is defined by
dαadxαaf(x)=f(α)a(x)=limϵ→0f(x+ϵ(x−a)1−α)−f(x)ϵ. | (1) |
If the limit (1) exist, then we say that f is left α -differentiable. Furthermore, if f(α)a(x) exist for x∈(a,b) , then
f(α)a(a)=limx→a+f(α)a(x) |
and
f(α)a(b)=limx→b−f(α)a(x). |
The right conformable fractional derivative of order α∈(0,1] terminating at b∈R of a function f:[a,b]→R is defined by
bdαbdxαf(x)=bf(α)(x)=−limϵ→0f(x+ϵ(b−x)1−α)−f(x)ϵ. | (2) |
If the limit (2) exist, then we say that f is right α -differentiable. Furthermore, if bf(α)(x) exist for x∈(a,b) , then
bf(α)(a)=limx→a+bf(α)(x) |
and
bf(α)(b)=limx→b−bf(α)(x). |
It is important to note that for α=1 the conformable fractional derivatives (1) and (2) reduce to first order ordinary derivatives. Furthermore, despite the definition of the conformable fractional derivatives (1) and (2) can be generalized for α>1 [6], we consider only 0<α≤1 in the present work. Is is also important to note that, differently from the majority of definitions of fractional derivative, including the popular Riemann-Liouville and Caputo fractional derivatives, the fractional derivatives (1) and (2) are local operators and are related to ordinary derivatives if the function is differentiable (see Remark 1). For more on local fractional derivatives, we refer the reader to [31], [32] and references therein.
Remark 1: If f∈C1[a,b] , then we have from (1) that
f(α)a(x)=(x−a)1−αf′(x) | (3) |
and from (2) that
bf(α)(x)=−(b−x)1−αf′(x) | (4) |
where f′(x) stands for the ordinary first order derivative of f(x) .
From (3) and (4) it is easy to see that the conformable fractional derivative of a constant is zero, differently from the Riemann-Liouville derivative of a constant, and for the power functions (x−a)p and (b−x)p one has
dαadxαa(x−a)p=p(x−a)p−α |
and
bdαbdxα(b−x)p=p(b−x)p−α |
for all p∈R .
The most remarkable consequence of definitions (1) and (2) is that the conformable fractional derivatives satisfy very simple fractional versions of chain and product rules.
Proposition 1 [5], [6]: Let 0<α<1 and f and g be α -differentiable functions. Then,
1)
(c1f+c2g)(α)a(x)=c1f(α)a(x)+c2g(α)a(x) |
and
b(c1f+c2g)(α)(x)=c1bf(α)(x)+c2bg(α)(x) |
for all c1,c2∈R .
2)
(fg)(α)a(x)=f(α)a(x)g(x)+f(x)g(α)a(x) |
and
b(fg)(α)(x)=bf(α)(x)g(x)+f(x)bg(α)(x). |
3)
(fg)(α)a(x)=f(α)a(x)g(x)−f(x)g(α)a(x)g2(x) |
and
b(fg)(α)(x)=bf(α)(x)g(x)−f(x)bg(α)(x)g2(x). |
4) If g(x)≥a , then
(f∘g)(α)a(x)=f(α)a(g(x))g(α)a(x)(g(x)−a)α−1. |
5) If g(x)≤b , then
b(f∘g)(α)(x)=bf(α)(g(x))bg(α)(x)(b−g(x))α−1. |
6) If g(x)<a , then
(f∘g)(α)a(x)=−af(α)(g(x))g(α)a(x)(a−g(x))α−1. |
7) If g(x)>b , then
b(f∘g)(α)(x)=−f(α)b(g(x))bg(α)(x)(g(x)−b)α−1. |
The simple chain and product rules given in Proposition 1 justify the increasing interest in the study of the conformable fractional calculus, since it enable us to investigate its potential applications as a tool to practical modeling of complex problems in science and engineering.
The conformable fractional integrals are defined as follows [5], [6].
Definition 2: The left conformable fractional integral of order 0<α≤1 starting from a∈R of a function f∈L1[a,b] is defined by
Iαaf(x)=∫xaf(u)dαau=∫xaf(u)(u−a)1−αdu | (5) |
and the right conformable fractional integral of order 0<α≤1 terminating at b∈R of a function f∈L1[a,b] is defined by
bIαf(x)=∫bxf(u)bdαu=∫bxf(u)(b−u)1−αdu. | (6) |
It is important to mention that the conformable fractional integrals (5) and (6) differ from the traditional fractional Riemann-Liouville integrals [1]-[3] only by a multiplicative constant. Moreover, for α=1 , the conformable fractional integrals reduce to ordinary first order integrals.
In addition to these definitions, in the present work we make use of the following properties of conformable fractional derivatives and integrals.
Theorem 1: Let f∈C[a,b] and 0<α≤1 . Then,
dαadxαaIαaf(x)=f(x) |
and
bdαbdxαbIαf(x)=f(x) |
for all x∈[a,b] .
Theorem 2 (Fundamental theorem of conformable fractional calculus): Let f∈C1[a,b] and 0<α≤1 . Then,
Iαaf(α)a(x)=f(x)−f(a) |
and
bIαbf(α)(x)=f(x)−f(b) |
for all x∈[a,b] .
Theorem 3 (Integration by parts): Let f,g:[a,b]→R be two functions such that fg is differentiable. Then,
∫baf(x)g(α)a(x)dαax=f(x)g(x)|ba−∫bag(x)f(α)a(x)dαax | (7) |
and
∫baf(x)bg(α)(x)bdαx=−f(x)g(x)|ba−∫bag(x)bf(α)(x)bdαx. | (8) |
If f,g:[a,b]→R are differentiable functions, then
∫baf(x)g(α)a(x)dαax=f(x)g(x)|ba+∫bag(x)bf(α)(x)bdαx. |
The proof of Theorem 1 follows directly from (3), (4), (5) and (6) since Iαaf(x) and bIαf(x) are differentiable. On the other hand, the fundamental theorem of the conformable fractional calculus (Theorem 2) is a direct consequence of (3), (4) and definitions (5) and (6) since f,g:[a,b]→R are differentiable functions. Finally, the integration by parts (7) and (8) follow from Proposition 1 and Theorem 1. We also need the following result.
Theorem 4 (Chain rule for functions of several variables): Let f:RN→R ( N∈N )} be a differentiable function in all its arguments and y1,...,yN:R→R be α -differentiable functions. Then,
dαadxαaf(y1(x),...,yN(x))=∂f∂y1y1(α)a+∂f∂y2y2(α)a+⋯+∂f∂yNyN(α)a | (9) |
and
bdαbdxαf(y1(x),...,yN(x))=∂f∂y1by1(α)+∂f∂y2by2(α)+⋯+∂f∂yNbyN(α). | (10) |
Proof : For simplicity, we prove (9) only for N=2 . The proofs for a general N and of (10) are similar. From (1) we have for N=2 that (by writing ¯x=x+ϵ(x−a)1−α for simplicity)
dαadxαaf(y1(x),y2(x))=limϵ→0f(y1(¯x),y2(¯x))−f(y1(x),y2(x))ϵ=limϵ→0f(y1(¯x),y2(¯x))−f(y1(x),y2(¯x))y1(¯x)−y1(x)y1(¯x)−y1(x)ϵ+limϵ→0f(y1(x),y2(¯x))−f(y1(x),y2(x))y2(¯x)−y2(x)y2(¯x)−y2(x)ϵ=∂f∂y1y1(α)a+∂f∂y2y2(α)a |
since f is differentiable.
Let us first consider the fractional variational integral
J(y)=∫baL(x,y(x),y(α)a(x))dαax | (11) |
defined on the set of continuous functions y:[a,b]→R such that y(α)a exists on [a,b] , where the Lagrangian L , L(x,y,y(α)a):[a,b]×R2→R , is of class C1 in each of its arguments. The fundamental problem of the calculus of variations consists in finding which functions extremize functional (11). In order to obtain a necessary condition for the extremum of (11) we need the following Lemma.
Lemma 1 (Fundamental lemma for the conformable calculus of variations): Let M and η be continuous functions on [a,b] .
If
∫baη(x)M(x)dαax=0 | (12) |
for any η∈C[a,b] with η(a)=η(b)=0 , then
M(x)=0 | (13) |
for all x∈[a,b] .
Proof : We do the proof by contradiction. From (12) we have that
∫baη(x)M(x)dαax=∫baη(x)M(x)(x−a)1−αdx=0. | (14) |
Suppose that there exist an x0∈(a,b) such that M(x0)≠0 . Without loss of generality, let us assume that M(x0)>0 . Since M is continuous on [a,b] , there exists a neighborhood Nδ(x0)⊂(a,b) such that
M(x)>0forallx∈Nδ(x0). |
Let us choose
η(x)={(x−x0−δ)2(x−x0+δ)2,if x∈Nδ(x0)0,if x∉Nδ(x0). | (15) |
Clearly, η(x) given by (15) is continuous and satisfy η(a)=0 and η(b)=0 . Inserting (15) into (14), we obtain that
∫baη(x)M(x)dαax=∫x0+δx0−δ(x−x0−δ)2(x−x0+δ)2M(x)(x−a)1−αdx>0 |
which contradicts our hypothesis. Thus,
M(x)(x−a)1−α>0forallx∈(a,b). |
Since (x−a)1−α>0 for x∈(a,b) , and since M∈C[a,b] , we get
M(x)=0forallx∈[a,b]. |
Theorem 5 (The conformable fractional Euler-Lagrange equation): Let J be a functional of form (11) with L∈C1 ([a,b]×R2) , and 0<α≤1 . Let y:[a,b]→R be a α -differentiable function with y(a)=ya and y(b)=yb , ya, yb∈R . Furthermore, let y∂L∂y(α)a be a differentiable function and ∂L∂y(α)a be α -differentiable. If y is an extremizer of J , then y satisfies the following fractional Euler-Lagrange equation:
∂L∂y−dαadxαa(∂L∂y(α)a)=0. | (16) |
Proof : Let y∗ give an extremum to (11). We define a family of functions
y(x)=y∗(x)+ϵη(x) | (17) |
where ϵ is a constant and η is an arbitrary α -differentiable function satisfying η∂L∂y∗(α)a∈C1 and the boundary conditions η(a)=η(b)=0 (weak variations). From (17), the boundary conditions η(a)=η(b)=0 , and the fact that y∗(a)=ya and y∗(b)=yb , it follows that function y is admissible: y is α -differentiable with y(a)=ya , y(b)=yb , and y∂L∂y∗(α)a is differentiable. Let the Lagrangian L be C1([a,b]×R2) . Because y∗ is an extremizer of functional J , the Gateaux derivative δJ(y∗) needs to be identically null. For the functional (11),
δJ(y∗)=limϵ→01ϵ(∫baL(x,y,y(α)a)dαax−∫baL(x,y∗,y∗(α)a)dαax)=∫ba(η(x)∂L(x,y∗,y∗(α)a)∂y∗+η(α)a(x)∂L(x,y∗,y∗(α)a)∂y∗(α)a)dαax=0. |
Using the integration by parts formula (7) ( η∂L∂y∗(α)a is differentiable), we get
δJ(y∗)=∫baη(x)(∂L(x,y∗,y∗(α)a)∂y∗−dαadxαa∂L(x,y∗,y∗(α)a)∂y∗(α)a)dαax=0 | (18) |
since η(a)=η(b)=0 . The fractional Euler-Lagrange equation (16) follows from (18) by using the fundamental Lemma 1.
Definition 3: A continuous function y solution of (16) is said to be an extremal of (11).
Remark 2: For α=1 , the functional J given by (11) reduces to the classical variational functional
J(y)=∫10L(x,y(x),y′(x))dx |
and the associated Euler-Lagrange equation (16) is
∂L∂y−ddx(∂L∂y′)=0. | (19) |
Let us now consider the more general case where the Lagrangian depends on both integer order and fractional order derivatives. In this case the following theorem holds.
Theorem 6 (The generalized conformable fractional Euler-Lagrange equation): Let J be a functional of form
J(y)=∫baL(x,y(x),y′(x),y(α)a(x))dx | (20) |
with L∈C1([a,b]×R3) , and 0<α≤1 . Let y:[a,b]→R be a differentiable function with y(a)=ya and y(b)=yb , ya,yb∈R . If y is an extremizer of J , then y satisfies the following fractional Euler-Lagrange equation:
∂L∂y−ddx(∂L∂y′)−1(x−a)1−αdαadxαa(∂˜L∂y(α)a)=0 | (21) |
where ˜L(x,y,y′,y(α)a)=(x−a)1−αL(x,y,y′,y(α)a) .
Proof : Let y∗ give an extremum to (20). We define a family of functions as in (17) but with y∈C1[a,b] . From (17) and the boundary conditions η(a)=η(b)=0 , and the fact that y∗(a)=ya and y∗(b)=yb , it follows that function y is admissible. Because y∗ is an extremizer of J , the Gateaux derivative δJ(y∗) needs to be identically null. For the functional (20) we have
δJ(y∗)=limϵ→01ϵ(∫baL(x,y,y′,y(α)a)dx−∫baL(x,y∗,y′∗,y∗(α)a)dx)=∫ba(η(x)∂L(x,y∗,y′∗,y∗(α)a)∂y∗+η′(x)∂L(x,y∗,y′∗,y∗(α)a)∂y′∗)dx+∫baη(α)a(x)∂L(x,y∗,y′∗,y∗(α)a)∂y∗(α)adx |
=∫baη(x)(∂L(x,y∗,y′∗,y∗(α)a)∂y∗−ddx∂L(x,y∗,y′∗,y∗(α)a)∂y′∗)dx+∫baη(α)a(x)∂˜L(x,y∗,y′∗,y∗(α)a)∂y∗(α)adαax=0 |
where we performed an integration by parts in the second term in the first integral (since η(a)=η(b)=0 ), and we rewrote the second integral as a conformable integral by using definition (5). Using the integration by parts formula (7) ( η∂L∂y∗(α)a is differentiable), we get
δJ(y∗)=∫baη(x)(∂L(x,y∗,y′∗,y∗(α)a)∂y∗−ddx∂L(x,y∗,y′∗,y∗(α)a)∂y′∗)dx−∫baη(x)dαadxαa∂˜L(x,y∗,y′∗,y∗(α)a)∂y∗(α)adαax=∫baη(x)((x−a)1−α∂L(x,y∗,y′∗,y∗(α)a)∂y∗−(x−a)1−αddx∂L(x,y∗,y′∗,y∗(α)a)∂y′∗−dαadxαa∂˜L(x,y∗,y′∗,y∗(α)a)∂y∗(α)a)dαax=0 | (22) |
since η(a)=η(b)=0 . The fractional Euler-Lagrange equation (21) follows from (22) by using the fundamental Lemma 1.
As an example of potential application of the variational calculus with conformable fractional derivatives, we formulate an action principle for dissipative systems free from the mathematical inconsistencies found in the Riewe approach [23] and far simpler than the formulation proposed in [23]. The action principle we propose states that the equation of motion for dissipative systems is obtained by taking the limit a→b in the extremal of the action
S=∫baL(x,x′,x(α)a)dt | (23) |
that satisfy the fractional Euler-Lagrange equation (see (21))
∂L∂x−ddt∂L∂x′−1(t−a)1−αdαadtαa∂˜L∂x(α)a=0 | (24) |
where ˜L(x,x′,x(α)a)=(t−a)1−αL(x,x′,x(α)a) , x(t) is the path of the particle and t is the time. It is important to emphasize that the condition a→b (also considered in the original Riewe's approach) applied to the action principle does not imply any restrictions for conservative systems, since in this case x(t) is the action's extremal for any time interval [a,b] , even when a→b . Furthermore, our action principle is simpler than the formulation in [23] and free from the mathematical inconsistencies present in Riewe's approach (see [23] for a detailed discussion). In order to show that our method provides us with physical Lagrangians, let us consider the simple problem of a particle under a frictional force proportional to velocity. A quadratic Lagrangian for a particle under a frictional force proportional to the velocity is given by
L(x,x′,x(12)a)=12m(x′)2−U(x)+γ2(x(12)a)2 | (25) |
where the three terms in (25) represent the kinetic energy, potential energy, and the fractional linear friction energy, respectively. Note that, differently from Riewe's Lagrangian [11], our Lagrangian (25) is a real function with a linear friction energy, which is physically meaningful. Since the equation of motion is obtained in the limit a→b , if we consider the last term in (25) up to first order in Δt=t−a , we get
γ2(x(12)a)2=γ2(x′Δt12)2≈γ2x′Δx |
that coincides, apart from the multiplicative constant 1/2, with the work from the frictional force γx′ in the displacement Δx≈x′Δt . The appearance of an additional multiplicative constant is a consequence of the use of fractional derivatives in the Lagrangian and does not appear in the equation of motion after we apply the action principle [23].
Remark 3: It is important to stress that the order of the fractional derivative should be fixed to α=1/2 in order to obtain, by a fractional Lagrangian, a correct equation of motion of a dissipative system. For α different from 1/2 , the Lagrangian does not describe a frictional system under a frictional force proportional to the velocity. Consequently, the fractional linear friction energy makes sense only for α=1/2 .
The Lagrangian (25) is physical in the sense it provides physically meaningful relations for the momentum and the Hamiltonian. If we define the canonical variables
q1=x′,q12=x(12)a |
and
p1=∂L∂q1=mx′,p12=∂L∂q12=γx(12)a |
we obtain the Hamiltonian
H=q1p1+q12p12−L=12m(x′)2+U(x)+γ2(x(12)a)2. | (26) |
From (26) we can see that the Lagrangian (25) is physical in the sense it provides us a correct relation for the momentum p1=m˙x and a physically meaningful Hamiltonian (it is the sum of all energies). Furthermore, the additional fractional momentum p12=γx(12)a goes to zero when we take the limit a→b , since x∈C2[a,b] .
Finally, the equation of motion for the particle is obtained by inserting our Lagrangian (25) into the Euler-Lagrange equation (24),
mx"+γ(t−a)−12d12adt12a[(t−a)12x(12)a]=mx"+γx′+γ(t−a)x"=F(x) | (27) |
where we have used (3) since x∈C2[a,b] and
F(x)=−ddxU(x) |
is the external force. By taking the limit a→b with t∈[a,b] , we finally obtain the correct equation of motion for a particle under a frictional force:
mx"+γx′=F(x). |
In the remainder of the present work, we are going to consider only the simplest case where we have no mixed integer and fractional derivatives. We now present the DuBois-Reymond condition in the conformable fractional context. It is an immediate consequence of the chain rule (9) and the Euler-Lagrange equation (16).
Theorem 7 (The conformable fractional DuBois-Reymond condition): If y is an extremal of J as in (11), then
dαadxαa(L−∂L∂y(α)ay(α)a)=∂L∂x⋅(x−a)1−α. | (28) |
Proof : By the chain rule (9) and the Leibniz rule in Proposition 1:
dαadxαa(L−∂L∂y(α)ay(α)a)=∂L∂xx(α)a+∂L∂yy(α)a+∂L∂y(α)adαadxαay(α)a−dαadxαa(∂L∂y(α)a)y(α)a−∂L∂y(α)adαadxαay(α)a=∂L∂xx(α)a+y(α)a[∂L∂y−dαadxαa(∂L∂y(α)a)]=∂L∂x⋅(x−a)1−α. |
Corollary 1: If (11) is autonomous, that is, if L=L(y,y(α)a) does not depend on x , then
dαadxαa(L−∂L∂y(α)ay(α)a)=0 |
along any extremal y .
Remark 4: When α=1 and y∈C1 , Theorem 7 is the classical DuBois-Reymond condition: if y∈C1 is an extremal of J(y)=∫10L(x,y,y′)dx (i.e., y satisfies (19)), then
ddx(L−∂L∂y′y′)=∂L∂x. |
We consider invariance transformations in the (x,y) -space, depending on a real parameter ϵ . To be more precise, we consider transformations of type
{¯x=x+ϵτ(x,y(x))¯y=y+ϵξ(x,y(x)) | (29) |
where the generators τ and ξ are such that ¯x≥a and there exist τ(α)a and ξ(α)a .
Definition 4: We say that the fractional variational integral (11) is invariant under the family of transformations (29) up to the Gauge term Λ , if a function Λ=Λ(x,y) exists such that for any function y and for any real x∈[a,b] , we have
L(¯x,¯y,dαa¯yd¯xαa)dαa¯xdαax=L(x,y,y(α)a)+ϵdαaΛdxαa(x,y)+o(ϵ) | (30) |
for all ϵ in some neighborhood of zero, where dαa¯xdαax stands for
dαa¯xdxαadαaxdxαa=1+ϵτ(α)a(x−a)1−α. | (31) |
We note that for α=1 our Definition 4 coincides with the standard approach (see, e.g., [33]). When Λ≡0 , one obtains the concept of absolute invariance. The presence of a new function Λ is due to the presence of external forces in the dynamical system, like friction. The function Λ is called a Gauge term. In fact, many phenomena are nonconservative and this has to be taken into account in the conservation laws [26], [34]. We give an example.
Example 1: Consider the transformation
{¯x=x¯y=y+ϵ12α(x−a)α | (32) |
and the functional
J(y)=∫ba(y(α)a(x))2dαax. | (33) |
Since
dαadxαa12α(x−a)α=12 |
it is easy to verify that (33) is invariant under (32) up to the Gauge function Λ=y .
Definition 5: Given a function C=C(x,y,y(α)a) , we say that C is a conserved quantity for (11) if
dαaCdxαa(x,y(x),y(α)a(x))=0 | (34) |
along any solution y of (16) (i.e., along any extremal of (11)).
Remark 5: Applying the conformable integral (5) to both sides of equation (34), Definition 5 is equivalent to C(x,y(x),y(α)a(x))≡const .
We now provide a necessary condition of invariance.
Theorem 8: If J given by (11) is invariant under a family of transformations (29), then
∂L∂xτ+∂L∂yξ+∂L∂y(α)a[ξ(α)a−y(α)a((α−1)τ(x−a)+τ(α)a(x−a)1−α)]+Lτ(α)a(x−a)1−α=dαaΛdxαa. | (35) |
Proof : By the fractional chain rule (see Proposition 1),
dαa¯yd¯xαa=dαa¯ydxαa(¯x−a)α−1dαa¯xdxαa=y(α)a+ϵξ(α)a(x+ϵτ−a)α−1[(x−a)1−α+ϵτ(α)a]. |
Substituting this formula into (30), differentiating with respect to ϵ and then putting ϵ=0 , we obtain relation (35).
Remark 6: Allowing α to be equal to 1, for Λ≡0 our equation (35) becomes the standard necessary condition of invariance (cf., e.g., [24]):
∂L∂xτ+∂L∂yξ+∂L∂y′(ξ′−y′τ′)+Lτ′=0. |
For α=1 and an arbitrary Λ , see [33].
In particular, if we consider "time invariance" (i.e., τ≡0 ), we obtain the following result.
Corollary 2: Let ¯y=y+ϵξ(x,y(x)) be a transformation that leaves invariant J in the sense that
L(x,¯y,¯y(α)a)=L(x,y,y(α)a)+ϵdαaΛdxαa(x,y)+o(ϵ). |
Then,
∂L∂yξ+∂L∂y(α)aξ(α)a=dαaΛdxαa. |
Noether's theorem is a beautiful result with important implications and applications in optimal control [35]-[37]. We provide here a conformable fractional Noether theorem in the context of the calculus of variations. Later, in Section Ⅸ, we provide a conformable fractional optimal control version (see Theorem 11).
Theorem 9 (The conformable fractional Noether theorem): If J given by (11) is invariant under (29) and if y is an extremal of J , then
dαadxαa[(L−∂L∂y(α)ay(α)a)τ+∂L∂y(α)aξ(x−a)1−α]=(1−α)∂L∂y(α)a[ξ(x−a)1−2α−y(α)aτ(x−a)α]+dαaΛdxαa(x−a)1−α. | (36) |
Proof : From Theorem 8, and using the conformable fractional Euler-Lagrange equation (16) and the DuBois-Reymond condition (28), we deduce successively that
dαaΛdxαa(x−a)1−α=[dαadxαa(L−∂L∂y(α)ay(α)a)τ(x−a)1−α+dαadxαa(∂L∂y(α)a)ξ+∂L∂y(α)aξ(α)a](x−a)1−α−∂L∂y(α)ay(α)a[(α−1)τ(x−a)α+τ(α)a)]+Lτ(α)a=[dαadxαa(L−∂L∂y(α)ay(α)a)τ+dαadxαa(∂L∂y(α)aξ)(x−a)1−α]−∂L∂y(α)ay(α)a[(α−1)τ(x−a)α+τ(α)a)]+Lτ(α)a=dαadxαa[(L−∂L∂y(α)ay(α)a)τ+∂L∂y(α)aξ(x−a)1−α]−∂L∂y(α)ay(α)a[(α−1)τ(x−a)α+τ(α)a]+Lτ(α)a−(L−∂L∂y(α)ay(α)a)τ(α)a−∂L∂y(α)aξ(1−α)(x−a)1−2α=dαadxαa[(L−∂L∂y(α)ay(α)a)τ+∂L∂y(α)aξ(x−a)1−α]+∂L∂y(α)ay(α)a(1−α)τ(x−a)α−∂L∂y(α)aξ(1−α)(x−a)1−2α. |
Thus, we obtain equation (36).
Remark 7: When \alpha=1 , equation (36) is simply Noether's conservation law in the presence of external forces: for any extremal of \mathcal{J} and for any family of transformations (\overline x, \overline y) for which \mathcal{J} is invariant, the conservation law
\left(L-\frac{\partial L}{\partial y'}y'\right)\tau + \frac{\partial L}{\partial y'}\xi =\Lambda+ {\rm constant} |
holds [33, Theorem 2.1]. In addition, if system is conservative ( \Lambda\equiv 0 ), then one has the classical Noether theorem
\left(L-\frac{\partial L}{\partial y'}y'\right)\tau + \frac{\partial L}{\partial y'}\xi ={\rm constant}. |
Corollary 3 (The conformable fractional Noether theorem under the presence of an external force f): If \mathcal{J} given by (11) is invariant under (29), y is an extremal of \mathcal{J} , and the function f=f (x, y, y^{(\alpha)}_a) satisfies the equation
\begin{equation*} \begin{split} \frac{d^\alpha_a f}{dx^\alpha_a} =& (1-\alpha)\frac{\partial L}{\partial y^{(\alpha)}_a}\left[\xi (x-a)^{1-2\alpha} -\frac{y^{(\alpha)}_a\tau}{(x-a)^{\alpha}}\right]\\ &+\frac{d^\alpha_a\Lambda}{dx^\alpha_a}(x-a)^{1-\alpha} \end{split} \end{equation*} |
then
\left(L-\frac{\partial L}{\partial y^{(\alpha)}_a}y^{(\alpha)}_a\right)\tau + \frac{\partial L}{\partial y^{(\alpha)}_a}\xi (x-a)^{1-\alpha}-f |
is a conserved quantity.
Corollary 4: If \mathcal{J} given by (11) is invariant under the transformation \overline x=x , \overline y=y+\epsilon \xi (x, y (x)) , and if y is an extremal of \mathcal{J} , then
\frac{\partial L}{\partial y^{(\alpha)}_a}\xi -\Lambda |
is a conserved quantity.
Proof : Given that
\frac{d^\alpha_a (x-a)^{1-\alpha}}{dx^\alpha_a} =(1-\alpha)(x-a)^{1-2\alpha} |
the result follows immediately from Theorem 9.
The Hamiltonian formalism is related to the Lagrangian one by the so called Legendre transformation, from coordinates and velocities to coordinates and momenta. Let the momenta be given by
\begin{equation} \label{defP} p(x)=\frac{\partial L}{\partial y^{(\alpha)}_a}(x, y(x), y^{(\alpha)}_a(x)) \end{equation} | (37) |
and the Hamiltonian function by
\begin{equation} \label{defH} H(x, y, v, \psi)=-L(x, y, v)+ \psi v. \end{equation} | (38) |
To simplify notation, [y](x) and \{y\}(x) will denote (x, y (x), y^{(\alpha)}_a (x)) and (x, y (x), y^{(\alpha)}_a (x), p (x)) , respectively. Differentiating (38), and using definition (37), it follows that
\begin{align} &\frac{d^\alpha_a H}{d x^\alpha_a}\{y\}(x)\\ &= -\frac{\partial L}{\partial x}[y](x)x^{(\alpha)}_a -\frac{\partial L}{\partial y}[y](x) \cdot y^{(\alpha)}_a(x)\\ &\quad-\frac{\partial L}{\partial v}[y](x) \cdot \frac{d^\alpha_a}{d x^\alpha_a}y^{(\alpha)}_a(x) +p^{(\alpha)}_a(x) \cdot y^{(\alpha)}_a(x)\\ &\quad + \frac{\partial L}{\partial v}[y](x) \cdot \frac{d^\alpha_a }{d x^\alpha_a}y^{(\alpha)}_a(x)\\ &=-\frac{\partial L}{\partial x}[y](x)\cdot (x-a)^{1-\alpha} -\frac{\partial L}{\partial y}[y](x) \cdot y^{(\alpha)}_a(x)\\ &\quad +p^{(\alpha)}_a (x)\cdot y^{(\alpha)}_a(x). \end{align} | (39) |
On the other hand, by the definition of Hamiltonian (38), one has immediately that
\begin{cases} \displaystyle \frac{\partial H}{\partial x}(x, y, v, \psi) =-\frac{\partial L}{\partial x}(x, y, v)\\[0.25cm] \displaystyle \frac{\partial H}{\partial y}(x, y, v, \psi) =-\frac{\partial L}{\partial y}(x, y, v)\\[0.25cm] \displaystyle \frac{\partial H}{\partial \psi}(x, y, v, \psi)=v \end{cases} |
and so we can write (39) in the form
\begin{align} \frac{d^\alpha_a H}{d x^\alpha_a}&\{y\}(x) \\=& \frac{\partial H}{\partial x}\{y\}(x)(x-a)^{1-\alpha}\\ & +\frac{\partial H}{\partial y}\{y\}(x) \cdot y^{(\alpha)}_a(x) +\frac{\partial H}{\partial \psi}\{y\}(x) \cdot p^{(\alpha)}_a(x). \end{align} | (40) |
If y is an extremal of \mathcal{J} , then by the conformable fractional Euler-Lagrange equation (16) one has
\frac{\partial L}{\partial y}[y](x) -\frac{d^\alpha_a}{d x^\alpha_a}\left(\frac{\partial L}{\partial v}[y]\right)(x) =-\frac{\partial H}{\partial y}\{y\}(x)-p^{(\alpha)}_a(x) = 0 |
and we can write
\begin{equation} \label{eq:EL:HF} \begin{cases} \displaystyle y^{(\alpha)}_a(x)=\frac{\partial H}{\partial \psi}\{y\}(x)\\[0.25cm] \displaystyle p^{(\alpha)}_a(x) =-\frac{\partial H}{\partial y}\{y\}(x). \end{cases} \end{equation} | (41) |
The system (41) is nothing else than the conformable fractional Euler-Lagrange equation in Hamiltonian form. Substituting the expressions of (41) into (40), we get the analog to the DuBois-Reymond condition (28) in Hamiltonian form:
\begin{equation} \label{eq:DR:Ham} \frac{d^\alpha_a H}{d x^{\alpha}_a}\{y\}(x) = \frac{\partial H}{\partial x}\{y\}(x)(x-a)^{1-\alpha}. \end{equation} | (42) |
If the Lagrangian L is autonomous, i.e., L does not depend on x , then
\frac{\partial L}{\partial x} = 0 |
and, consequently, by (42) H is a conserved quantity. If the Lagrangian L does not depend on y , then
\frac{\partial L}{\partial y}=- \frac{\partial H}{\partial y}= 0 |
and so p^{(\alpha)}_a=0 , i.e., p is a conserved quantity.
We now exhibit Corollary 3 within the Hamiltonian framework.
Theorem 10 (Conformable fractional Noether's theorem in Hamiltonian form under the presence of an external force f): If \mathcal{J} given by (11) is invariant under (29), y is an extremal of \mathcal{J} , and function f=f (x, y (x), y^{(\alpha)}_a (x)) satisfies the equation
\begin{equation*} \begin{split} &\frac{d^\alpha_a f}{dx^\alpha_a}(x, y(x), y^{(\alpha)}_a(x))\\ &\qquad =(1-\alpha)p(x)\left[\xi (x-a)^{1-2\alpha} -\frac{y^{(\alpha)}_a(x)\tau}{(x-a)^\alpha}\right]\\ &\qquad \quad+\frac{d^\alpha_a\Lambda}{dx^\alpha_a}(x, y(x))(x-a)^{1-\alpha} \end{split} \end{equation*} |
then
p(x)\xi (x-a)^{1-\alpha}-H\{y\}(x)\tau-f(x, y(x), y^{(\alpha)}_a(x)) |
is a conserved quantity.
The conformable fractional optimal control problem is stated as follows: find a pair of functions (y (\cdot), v (\cdot)) that minimizes
\begin{equation} \label{functOptimal} \mathcal{J}(y, v)=\int_a^b L(x, y(x), v(x)) d^\alpha_ax \end{equation} | (43) |
when subject to the (nonautonomous) fractional control system
\begin{equation} \label{contraintOptimal} y^{(\alpha)}_a(x)=\varphi (x, y(x), v(x)). \end{equation} | (44) |
A pair (y (\cdot), v (\cdot)) that minimizes functional (43) subject to (44) is called an optimal process. The reader interested on the fractional optimal control theory is referred to [28], [29], [38]. Here we note that if \alpha=1 , then (43) and (44) is the standard optimal control problem: to minimize
\mathcal{J}(y, v)=\int_a^b L(x, y(x), v(x)) dx |
subject to the control system
y'(x)=\varphi (x, y(x), v(x)). |
We assume that the Lagrangian L and the velocity vector \varphi are functions at least of class C^1 in their domain [a, b]\times {\mathbb R}^2 . Also, the admissible state trajectories y are such that y^{(\alpha)}_a exist.
Remark 8: In case \varphi \equiv v , the previous problem (43) and (44) reduces to the fundamental problem of the conformable fractional variational calculus (11), as stated in Section Ⅲ.
Following the standard approach [37], [39], we consider the augmented conformable fractional functional
\begin{equation} \label{augmfunction} \begin{split} \mathcal{I}(y, v, p)=\int_a^b [& L(x, y(x), v(x))+p(x)(y^{(\alpha)}_a(x)\\ &-\varphi (x, y(x), v(x)))] d^\alpha_a x \end{split} \end{equation} | (45) |
where p is such that p^{(\alpha)}_a exists. Consider a variation vector of type (y+\epsilon y_1, v+\epsilon v_1, p+\epsilon p_1) with |\epsilon| \ll 1 . For convenience, we restrict ourselves to the case y_1(a)=y_1(b)=0 . If (y (\cdot), v (\cdot)) is an optimal process, then the first variation is zero when \epsilon=0 . Thus, using the conformable fractional integration by parts formula (Theorem 3), we obtain that
\begin{array}{ll} 0\!\!\!&=\displaystyle\int_a^b \Biggl[\frac{\partial L}{\partial y}y_1 + \frac{\partial L}{\partial v}v_1 +p_1 (y^{(\alpha)}_a-\varphi)\\ &\quad\qquad +p\left({y_1}^{(\alpha)}_a-\frac{\partial\varphi}{\partial y}y_1 -\frac{\partial\varphi}{\partial v}v_1\right) \Biggr] d^\alpha_a x\\ &=\displaystyle\int_a^b \Biggl[y_1 \left( \frac{\partial L}{\partial y}-p \frac{\partial\varphi}{\partial y}- p^{(\alpha)}_a \right) +v_1\left( \frac{\partial L}{\partial v} -p \frac{\partial\varphi}{\partial v} \right)\\ &\displaystyle \quad\qquad + p_1 (y^{(\alpha)}_a-\varphi) \Biggr] d^\alpha_a x. \end{array} |
By the arbitrariness of the the variation functions, we obtain the following system, called the Euler-Lagrange equations for the conformable fractional optimal control problem:
\begin{equation} \label{optimalsystem} \left\{\begin{array}{l} y^{(\alpha)}_a(x)=\varphi(x, y(x), v(x))\\ p^{(\alpha)}_a(x)=\displaystyle\frac{\partial L}{\partial y}(x, y(x), v(x)) -p(x) \frac{\partial\varphi}{\partial y}(x, y(x), v(x))\\ \displaystyle\frac{\partial L}{\partial v}(x, y(x), v(x)) -p (x)\frac{\partial\varphi}{\partial v}(x, y(x), v(x)) =0. \end{array}\right. \end{equation} | (46) |
These equations give necessary conditions for finding the optimal solutions of problem (43) and (44). We remark that they are similar to the standard ones, in case of integer order derivatives, but in this case they contain conformable fractional derivatives, as expected. The solution can be stated using the Hamiltonian formalism. Consider the Hamiltonian function
\begin{equation} \label{hamilDef} H(x, y, v, p)=-L(x, y, v)+p(x) \varphi(x, y, v). \end{equation} | (47) |
Then (46) gives:
1) The fractional Hamiltonian system
\begin{equation} \label{Hamil1} \left\{ \begin{array}{l} \displaystyle y^{(\alpha)}_a(x)=\frac{\partial H}{\partial p}(x, y, v, p)\\[0.3cm] \displaystyle p^{(\alpha)}_a(x)=-\frac{\partial H}{\partial y}(x, y, v, p).\\ \end{array} \right. \end{equation} | (48) |
2) The stationary condition
\begin{equation} \label{Hamil2} \frac{\partial H}{\partial v}(x, y, v, p)=0. \end{equation} | (49) |
Definition 6: Any triplet (y, v, p) satisfying system (48) and (49) is called a conformable fractional Pontryagin extremal.
Remark 9: In the particular case \varphi \equiv v , i.e., when the conformable fractional optimal control problem is reduced to the fundamental conformable fractional problem of the calculus of variations, we obtain
H=-L(x, y, v)+pv, \quad y^{(\alpha)}_a=v |
and the equations
p^{(\alpha)}_a=-\frac{\partial H}{\partial y} =\frac{\partial L}{\partial y}, \quad p=\frac{\partial L}{\partial v}. |
Therefore, we obtain the conformable fractional Euler-Lagrange equation (16):
\frac{\partial L}{\partial y}=\frac{d^\alpha_a}{dx^\alpha_a}\left( \frac{\partial L}{\partial y^{(\alpha)}_a}\right). |
Let us now considerer the augmented fractional variational functional (45) written in the Hamiltonian form:
\begin{equation} \label{fafh} \mathcal{I}(y, v, p)=\int_0^1 (-H(x, y(x), v(x), p(x)) +p(x)y^{(\alpha)}_a(x)) d^\alpha_a x \end{equation} | (50) |
where H is given by expression (47). For a parameter \epsilon , with |\epsilon| \ll 1 , consider the family of transformations
\begin{equation} \label{trans4} ~~~~~~~~~~~~~~~~~~~~~~~~\left\{ \begin{array}{l} \overline x=x+\epsilon \tau (x, y(x), v(x), p(x))~~~~~~~\\ \overline y=y+\epsilon \xi (x, y(x), v(x), p(x))~~~~~~~\\ \overline v=v+\epsilon \sigma (x, y(x), v(x), p(x))~~~~~~~\\ \overline p=p+\epsilon \pi (x, y(x), v(x), p(x)). \end{array}\right. \end{equation} | (51) |
We now define the notion of invariance of (43)-(44) in terms of the Hamiltonian H and the augmented conformable fractional variational functional (50).
Definition 7: The conformable fractional optimal control problem (43) and (44) is invariant under the transformations (51) up to the Gauge term \Lambda , if a function \Lambda=\Lambda (x, y) exists such that for any functions y, v and p , and for any real x\in[0,1] , the following equality holds:
\begin{align} &\left[-H\left(\overline x, \overline y, \overline v, \overline p\right) +\overline p \frac{d^\alpha_a\overline y}{d\overline x^\alpha_a}\right] \frac{d^\alpha_a\overline x}{d^\alpha_a x}\\ &\qquad =-H(x, y, v, p)+py^{(\alpha)}_a+\epsilon \frac{d^\alpha_a\Lambda}{dx^\alpha_a}(x, y)+ o(\epsilon) \end{align} | (52) |
for all \epsilon in some neighborhood of zero, where as in Definition 4 \frac{d^\alpha_a\overline x}{d^\alpha_a x} stands for (31).
Theorem 11 (Fractional Noether's theorem for the fractional optimal control problem (43)-(44)): If (43) and (44) is invariant under (51) in the sense of Definition 7, and if (y, v, p) is a conformable fractional Pontryagin extremal, then
\begin{align} \frac{d^\alpha_a}{dx^\alpha_a}(p\xi)&-\tau\left(\frac{\partial H}{\partial x} +(\alpha-1)\frac{p y^{(\alpha)}_a}{x-a} \right)\\ &-H\frac{\tau^{(\alpha)}_a}{(x-a)^{1-\alpha}}=\frac{d^\alpha_a\Lambda}{dx^\alpha_a}. \end{align} | (53) |
Proof : Differentiating (52) with respect to \epsilon , then choosing \epsilon=0 , we get
\begin{align*} &-\frac{\partial H}{\partial x}\tau-\frac{\partial H}{\partial y}\xi -\frac{\partial H}{\partial v}\sigma -\frac{\partial H}{\partial p}\pi+\pi y^{(\alpha)}_a\\ &+p\left[ \xi^{(\alpha)}_a-y^{(\alpha)}_a\left((\alpha-1)\frac{\tau}{x-a} +\frac{\tau^{(\alpha)}_a}{(x-a)^{1-\alpha}}\right) \right]\\ &+\left[-H+py^{(\alpha)}_a\right]\frac{\tau^{(\alpha)}_a}{(x-a)^{1-\alpha}} =\frac{d^\alpha_a\Lambda}{dx^\alpha_a}. \end{align*} |
Equation (53) follows because (y, v, p) is a conformable fractional Pontryagin extremal.
Remark 10: When \alpha=1 and \Lambda=0 , equation (53) becomes
\frac{d}{dx}(p\xi)-\tau\frac{\partial H}{\partial x}-H\tau'=0. |
Using relations (48) and (49) with \alpha=1 , we deduce that
-H\tau + p\xi \equiv {\rm{constant}} |
which is the optimal control version of Noether's theorem [35]-[37]. For \alpha\in (0, 1) , Theorem 11 extends the main result of [28].
In this section, we show a necessary condition of invariance, when the Lagrangian depends on two independent variables x_1 and x_2 and on m functions y_1, ..., y_m . First, we define conformable fractional partial derivatives and conformable multiple fractional integrals in a natural way, similarly as done in the integer case. In addition, we are going to use the following properties.
Theorem 12 (Conformable Green's theorem for a rectangle): Let f and g be two continuous and \alpha -differentiable functions whose domains contain R=[a, b]\times[c, d]\subset {\mathbb R}^2 . Then,
\begin{align} \label{green} \int_a^b&\left(f(x_1, c)\!-\!f(x_1, d)\right)d^\alpha_ax_1 \!+\!\int_c^d\left(g(b, x_2)\!-\!g(a, x_2)\right)d^\alpha_cx_2\\ &=\int_R \left(\frac{\partial ^\alpha_a }{\partial {x_1}^\alpha_a} g(x_1, x_2) -\frac{\partial ^\alpha_c }{\partial {x_2}^\alpha_c} f(x_1, x_2)\right) d^\alpha_a x_1d^\alpha_c x_2. \end{align} | (54) |
Proof : By Theorem 2, we have
\begin{equation*} f(x_1, d)-f(x_1, c) =\int_c^d\frac{\partial ^\alpha_c }{\partial {x_2}^\alpha_c} f(x_1, x_2)d^\alpha_cx_2 \end{equation*} |
and
\begin{equation*} g(b, x_2)-g(a, x_2) =\int_a^b\frac{\partial ^\alpha_a }{\partial {x_1}^\alpha_a} g(x_1, x_2)d^\alpha_ax_1. \end{equation*} |
Therefore,
\begin{equation*} \begin{split} \int_a^b\!\! &\left(f(x_1, c)\!-\!f(x_1, d)\right)d^\alpha_ax_1 \! +\!\int_c^d\!\!\left(g(b, x_2)\!-\!g(a, x_2)\right)d^\alpha_cx_2\\ &=-\int_a^b\int_c^d\frac{\partial ^\alpha_c }{\partial {x_2}^\alpha_c} f(x_1, x_2)d^\alpha_cx_2d^\alpha_ax_1\\ &\quad +\int_c^d\int_a^b \frac{\partial ^\alpha_a }{\partial {x_1}^\alpha_a} g(x_1, x_2)d^\alpha_ax_1d^\alpha_cx_2\\ &=\int_R \left(\frac{\partial ^\alpha_a }{\partial {x_1}^\alpha_a} g(x_1, x_2) -\frac{\partial ^\alpha_c }{\partial {x_2}^\alpha_c} f(x_1, x_2)\right)d^\alpha_a x_1d^\alpha_c x_2. \end{split} \end{equation*} |
Remark 11: From Definition 2 and Remark 1, it is easy to verify that for C^1 functions our fractional Green's theorem over a rectangular domain (Theorem 12) reduces to the conventional Green's identity for
\tilde{f}(x_1, x_2)=f(x_1, x_2)(x_1-a)^{\alpha-1} |
and
\tilde{g}(x_1, x_2)=g(x_1, x_2)(x_2-a)^{\alpha-1}. |
Lemma 2: Let f, G and h be \alpha -differentiable continuous functions whose domains contain R=[a, b]\times[c, d] . If h=0 on the boundary \partial R of R , then
\begin{align} \label{lemma1} \int_R &\left(G (x_1, x_2)\frac{\partial ^\alpha_a }{\partial {x_1}^\alpha_a} h(x_1, x_2)\right.\\ &\quad\left.-F(x_1, x_2)\frac{\partial ^\alpha_c }{\partial {x_2}^\alpha_c} h(x_1, x_2)\right)d^\alpha_a x_1d^\alpha_c x_2\\ &=-\int_R\left(\frac{\partial ^\alpha_a }{\partial {x_1}^\alpha_a} G(x_1, x_2) -\frac{\partial ^\alpha_c }{\partial {x_2}^\alpha_c} F(x_1, x_2) \right)\\ &\quad\times h(x_1, x_2)d^\alpha_a x_1d^\alpha_c x_2. \end{align} | (55) |
Proof : By choosing f=Fh and g=Gh in Green's formula (54), we obtain that
\begin{align*}&\int_a^b \left(F(x_1, c)h(x_1, c)-F(x_1, d)h(x_1, d)\right)d^\alpha_ax_1\\ &+\int_c^d\left(G(b, x_2)g(b, x_2)-G(a, x_2)h(a, x_2)\right)d^\alpha_cx_2 \end{align*} |
\begin{equation*} \begin{split} \!\!\!\!=&\int_R \left(\frac{\partial ^\alpha_a }{\partial {x_1}^\alpha_a} G(x_1, x_2)\right.\\ & \qquad\left.-\frac{\partial ^\alpha_c }{\partial {x_2}^\alpha_c} F(x_1, x_2)\right)h(x_1, x_2)d^\alpha_a x_1d^\alpha_c x_2\\ & +\int_R \left(G(x_1, x_2)\frac{\partial^\alpha_a }{\partial {x_1}^\alpha_a} h(x_1, x_2) \right.\\ & \quad\qquad \left.-F(x_1, x_2)\frac{\partial^\alpha_c}{\partial {x_2}^\alpha_c} h(x_1, x_2)\right)d^\alpha_a x_1d^\alpha_c x_2. \end{split} \end{equation*} |
Since h=0 on the boundary \partial R of R , we have
\begin{align*} &\int_R \left(G(x_1, x_2)\frac{\partial^\alpha_a}{\partial {x_1}^\alpha_a} h(x_1, x_2)\right. \\ &\quad\quad\left.-F(x_1, x_2)\frac{\partial^\alpha_c}{\partial {x_2}^\alpha_c} h(x_1, x_2)\right)d^\alpha_a x_1d^\alpha_c x_2\\ =&-\int_R \left(\frac{\partial^\alpha_a}{\partial {x_1}^\alpha_a} G(x_1, x_2) -\frac{\partial ^\alpha_c }{\partial {x_2}^\alpha_c} F(x_1, x_2)\right)\\ &\quad\quad\quad\times h(x_1, x_2)d^\alpha_a x_1d^\alpha_c x_2. \end{align*} |
Remark 12: In the very recent and general paper [40], a vector calculus with deformed derivatives (as the conformable derivative) is formally introduced. We refer the reader to [40] for a detailed discussion of a vector calculus with deformed derivatives and more properties on the multi-dimensional conformable calculus.
Let us now consider the fractional variational integral
\begin{equation} \label{funct2} \mathcal{J}(y) =\int_R L\left(x, y, \frac{\partial^\alpha_a y}{\partial x^\alpha_a}\right)d^\alpha_a x \end{equation} | (56) |
where for simplicity we choose R=[a, b] \times [a, b] , and where x=(x_1, x_2) , y=(y_1, ..., y_m) , d^\alpha_a x=d^\alpha_a x_1d^\alpha_a x_2 , and
\frac{\partial ^\alpha_a y}{\partial x^\alpha_a} =\left( \frac{\partial ^\alpha_a y_1}{\partial {x_1}^\alpha_a}, \frac{\partial ^\alpha_a y_1}{\partial {x_2}^\alpha_a}, ..., \frac{\partial ^\alpha_a y_m}{\partial {x_1}^\alpha_a}, \frac{\partial ^\alpha_a y_m}{\partial {x_2}^\alpha_a} \right). |
We are assuming that the Lagrangian
L=L(x_1, x_2, y_1, ..., y_m, v_{1, 1}, v_{1, 2}, ..., v_{m, 1}, v_{m, 2}) |
is at least of class C^1 , that the domains of y_k , k\in\{1, ..., m\} , contain R , and that all these partial conformable fractional derivatives exist.
Theorem 13 (The multi-dimensional fractional Euler-Lagrange equation): Let y be an extremizer of (56) with
y|_{\partial R}=\psi(x_1, x_2) |
for some given function \psi=(\psi_1, ..., \psi_m) . Then, the following equation holds:
\begin{equation} \label{FracELEquation2} \frac{\partial L}{\partial y_k}-\frac{\partial^\alpha_a}{\partial {x_1}^\alpha_a}\left( \frac{\partial L}{\partial v_{k, 1}}\right)-\frac{\partial^\alpha_a}{\partial {x_2}^\alpha_a}\left( \frac{\partial L}{\partial v_{k, 2}}\right)=0 \end{equation} | (57) |
for all k\in\{1, ..., m\} .
Proof : Let y^{\ast}=(y_1^{\ast}, ..., y_m^{\ast}) give an extremum to (56). We define m families of functions
\begin{equation} \label{proof2} y_k(x_1, x_2)=y_k^{\ast}(x_1, x_2)+\epsilon\eta_k(x_1, x_2) \end{equation} | (58) |
where k\in \{1, ..., m\} , \epsilon is a constant, and \eta_k is an arbitrary \alpha -differentiable function satisfying the boundary conditions \eta_k|_{\partial R}=0 (weak variations). From (58), the boundary conditions \eta_k|_{\partial R}=0 and y_k|_{\partial R}=\psi_k (x_1, x_2) , it follows that function y_k is admissible. Let the Lagrangian L be C^1 . Because y^{\ast} is an extremizer of functional \mathcal{J} , the Gateaux derivative \delta \mathcal{J}(y^{\ast}) needs to be identically null. For the functional (56),
\begin{equation*} \begin{split} \delta \mathcal{J}(y^{\ast}) &=\lim_{\epsilon\rightarrow 0} \frac{1}{\epsilon}\Bigg( \int_R L\left(x, y, \frac{\partial^\alpha_a y}{\partial x^\alpha_a}\right) d^\alpha_ax \Bigg.\\ &\qquad\qquad\Bigg.-\int_R L\left(x, y^{\ast}, \frac{\partial^\alpha_a y^{\ast}}{\partial x^\alpha_a}\right) d^\alpha_ax\Bigg)\\ &=\sum_{k=1}^m\int_R \left(\eta_k(x_1, x_2)\displaystyle \frac{\partial L\left(x, y^{\ast}, \displaystyle \frac{\partial^\alpha_a y^{\ast}}{\partial x^\alpha_a}\right)}{\partial y_k^{\ast}}\right.\\ &~~~~ +\frac{\partial^\alpha_a}{\partial {x_1}^\alpha_a}\eta_k(x_1, x_2) \frac{\partial L\left(x, y^{\ast}, \displaystyle \frac{\partial^\alpha_a y^{\ast}}{\partial x^\alpha_a}\right)}{\partial v_{k, 1}} \\ &~~~~\left.+\frac{\partial^\alpha_a}{\partial {x_2}^\alpha_a}\eta_k(x_1, x_2)\frac{\partial L\left(x, y^{\ast}, \displaystyle \frac{\partial^\alpha_a y^{\ast}}{\partial x^\alpha_a}\right)}{\partial v_{k, 2}}\right)d^\alpha_ax\\& =0. \end{split} \end{equation*} |
Using (55), we get that
\begin{equation} \label{proof4} \begin{split} &\sum_{k=1}^m\int_R \eta_k(x_1, x_2)\left(\frac{\partial L\left(x, y^{\ast}, \displaystyle \frac{\partial^\alpha_a y^{\ast}}{\partial x^\alpha_a}\right)}{\partial y_k^{\ast}}\right.\\ &\qquad \quad\qquad-\frac{\partial^\alpha_a}{\partial {x_1}^\alpha_a}\frac{\partial L\left(x, y^{\ast}, \displaystyle \frac{\partial^\alpha_a y^{\ast}}{\partial x^\alpha_a}\right)}{\partial v_{k, 1}}~~~~~~~~\\ &\qquad \quad\qquad\left.-\frac{\partial^\alpha_a}{\partial {x_2}^\alpha_a}\frac{\partial L\left(x, y^{\ast}, \displaystyle \frac{\partial^\alpha_a y^{\ast}}{\partial x^\alpha_a}\right)}{\partial v_{k, 2}} \right)d^\alpha_ax=0 \end{split} \end{equation} | (59) |
since \eta_k|_{\partial R}=0 . The fractional Euler-Lagrange equation (57) follows from (59) by using the fundamental lemma of the conformable fractional calculus of variations (Lemma 1).
Let \epsilon be a real, and consider the following family of transformations:
\begin{equation} \label{trans2} \left\{ \begin{array}{ll} \overline x_i=x_i+\epsilon \tau_i (x, y(x)), & i \in \{1, 2\}, \\ \overline y_k=y_k+\epsilon \xi_k (x, y(x)), & k \in \{1, ..., m\} \end{array} \right. \end{equation} | (60) |
where \tau_i and \xi_k are such that there exist \frac{\partial^\alpha_a\tau_i}{\partial {x_j}^\alpha_a} and \frac{\partial ^\alpha_a\xi_k}{\partial {x_j}^\alpha_a} for all i, j \in \{1, 2\} and all k \in \{1, ..., m\} . Denote by \left[\frac{\partial^\alpha_a \overline x}{\partial^\alpha_a x} \right] the matrix
\begin{align*} \displaystyle &\left[ \begin{array}{cc} \displaystyle\dfrac {\dfrac{\partial^\alpha_a \overline x_1}{\partial{x_1}^\alpha_a}} {\dfrac{\partial^\alpha_a x_1}{\partial {x_1}^\alpha_a}} &\displaystyle\dfrac {\dfrac{\partial^\alpha_a \overline x_1}{\partial{x_2}^\alpha_a}} {\dfrac{\partial^\alpha_a x_2}{\partial {x_2}^\alpha_a}}\\[9mm] \displaystyle\dfrac {\dfrac{\partial^\alpha_a \overline x_2}{\partial{x_1}^\alpha_a}} {\dfrac{\partial^\alpha_a x_1}{\partial {x_1}^\alpha_a}} &\displaystyle\dfrac {\dfrac{\partial^\alpha_a \overline x_2}{\partial{x_2}^\alpha_a}} {\dfrac{\partial^\alpha_a x_2}{\partial {x_2}^\alpha_a}} \end{array} \right]\\\end{align*} |
\begin{align*} = \displaystyle \left[ \begin{array}{cc} \displaystyle 1+\frac{\epsilon}{(x_1-a)^{1-\alpha}} \frac{\partial^\alpha_a\tau_1}{\partial {x_1}^\alpha_a} & \displaystyle\frac{\epsilon}{(x_2-a)^{1-\alpha}} \frac{\partial^\alpha_a\tau_1}{\partial {x_2}^\alpha_a}\\[4mm] \displaystyle\frac{\epsilon}{(x_1-a)^{1-\alpha}} \frac{\partial^\alpha_a\tau_2}{\partial {x_1}^\alpha_a} &\displaystyle 1+\frac{\epsilon}{(x_2-a)^{1-\alpha}} \frac{\partial^\alpha_a\tau_2}{\partial {x_2}^\alpha_a} \end{array} \right]. \end{align*} |
Definition 8: Functional \mathcal{J} as in (56) is invariant under the family of transformation (60) if for all y_k and for all x_i\in[0,1] we have
\begin{multline*} \qquad L\left(\overline x, \overline y, \frac{\partial^\alpha_a\overline y}{ \partial\overline x^\alpha_a}\right) \det \left[\frac{\partial^\alpha_a \overline x}{\partial^\alpha_a x} \right]\\ =L\left(x, y, \frac{\partial^\alpha_a y}{\partial x^\alpha_a}\right) +\epsilon \frac{d^\alpha_a\Lambda}{dx^\alpha_a}(x, y)+o(\epsilon)\qquad\qquad \end{multline*} |
for all \epsilon in some neighborhood of zero.
Using the same techniques as in the proof of Theorem 8, we obtain a necessary condition of invariance for the fractional variational problem (56).
Theorem 14: If \mathcal{J} given by (56) is invariant under transformations (60), then
\begin{align} \label{eq:cni:md} &\sum_{i=1}^2 \frac{\partial L}{\partial x_i}\tau_i + \sum_{k=1}^m \frac{\partial L}{\partial y_k}\xi_k + \sum_{k=1}^m \sum_{i=1}^2\frac{\partial L}{\partial v_{k, i}}\left[ \frac{\partial ^\alpha_a \xi_k}{\partial {x_i}^\alpha_a}\right.\\ &\left.-\frac{\partial ^\alpha_a y_k}{\partial {x_i}^\alpha_a}\left( (\alpha-1)\frac{\tau_i}{x_i-a}+\frac{1}{(x_i-a)^{1-\alpha}} \frac{\partial^\alpha_a \tau_i }{\partial {x_i}^\alpha_a} \right) \right]\\ &+ L\left( \frac{1}{(x_1-a)^{1-\alpha}} \frac{\partial^\alpha_a\tau_1}{\partial {x_1}^\alpha_a} + \frac{1}{(x_2-a)^{1-\alpha}} \frac{\partial^\alpha_a\tau_2}{\partial {x_2}^\alpha_a}\right) =\frac{d^\alpha\Lambda}{dx^\alpha}. \end{align} | (61) |
Proof : Using relations
\displaystyle\frac{\partial^\alpha_a\overline y_k}{ \partial{{\overline x}_i}^\alpha_a} =\frac{\displaystyle \frac{\partial^\alpha_a y_k}{\partial {x_i}^\alpha_a} +\epsilon\displaystyle \frac{\partial^\alpha_a \xi_k}{\partial {x_i}^\alpha_a} }{(x_i +\epsilon\tau_i-a)^{\alpha-1}\left[(x_i-a)^{1-\alpha} +\epsilon \displaystyle \frac{\partial^\alpha_a \tau_i }{\partial {x_i}^\alpha_a} \right]} |
and
\begin{equation*} \begin{split} \displaystyle &\left.\frac{d}{d\epsilon}\det \left[ \frac{\partial^\alpha \overline x}{\partial^\alpha x} \right]\right|_{\epsilon=0}\\ &\qquad =\frac{1}{(x_1-a)^{1-\alpha}} \frac{\partial^\alpha_a\tau_1}{\partial {x_1}^\alpha_a} + \frac{1}{(x_2-a)^{1-\alpha}} \frac{\partial^\alpha_a\tau_2}{\partial {x_2}^\alpha_a}~~~~~~ \end{split} \end{equation*} |
we conclude that (61) holds.
Remark 13: When \alpha=1 and \Lambda\equiv0 , Theorem 14 reduces to the standard one [24]: equality (61) simplifies to
\begin{equation*} \begin{split} \sum_{i=1}^2 \frac{\partial L}{\partial x_i}\tau_i& + \sum_{k=1}^m \frac{\partial L}{\partial y_k}\xi_k + \sum_{k=1}^m \sum_{i=1}^2\frac{\partial L}{\partial v_{k, i}}\left[ \frac{\partial \xi_k}{\partial x_i}-\frac{\partial y_k}{\partial x_i} \frac{\partial \tau_i }{\partial x_i}\right]\\ & +L\left(\frac{\partial\tau_1}{\partial x_1} + \frac{\partial\tau_2}{\partial x_2}\right)=0. \end{split} \end{equation*} |
Corollary 5: If \mathcal{J} given by (56) is invariant under (60), \tau_1\equiv0\equiv\tau_2 , and no Gauge term is involved (i.e., \Lambda\equiv0 ), then
\sum_{k=1}^m \frac{\partial L}{\partial y_k}\xi_k + \sum_{k=1}^m \sum_{i=1}^2\frac{\partial L}{\partial v_{k, i}}\frac{\partial ^\alpha_a \xi_k}{\partial {x_i}^\alpha_a}=0. |
It remains an open question how to obtain a Noether constant of motion for the conformable fractional multi-dimensional case.
[1] |
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies). Amsterdam: Elsevier Science, 2006.
|
[2] |
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley-Interscience, 1993.
|
[3] |
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering. San Diego, CA: Academic Press, Inc., 1999.
|
[4] |
J. T. Machado, V. Kiryakova, and F. Mainardi, "Recent history of fractional calculus, " Commun. Nonlinear Sci. Numer. Simul. , vol. 16, no. 3, pp. 1140-1153, Mar. 2011. http://www.sciencedirect.com/science/article/pii/S1007570410003205
|
[5] |
R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, "A new definition of fractional derivative, " J. Comput. Appl. Math. , vol. 264, pp. 65-70, Jul. 2014. http://www.sciencedirect.com/science/article/pii/S0377042714000065
|
[6] |
T. Abdeljawad, "On conformable fractional calculus, " J. Comput. Appl. Math. , vol. 279, pp. 57-66, May2015. http://www.sciencedirect.com/science/article/pii/S0377042714004622
|
[7] |
D. R. Anderson and R. I. Avery, "Fractional-order boundary value problem with Sturm-Liouville boundary conditions, " Electron. J. Diff. Equ. , vol. 2015, no. 29, pp. 10, Jan. 2015. http://www.wenkuxiazai.com/doc/6d8cb0076c85ec3a87c2c5c2.html
|
[8] |
H. Batarfi, J. Losada, J. J. Nieto, and W. Shammakh, "Three-point boundary value problems for conformable fractional differential equations, " J. Funct. Spaces, vol. 2015, Article ID 706383, 2015. http://www.sciencedirect.com/science/article/pii/S0898122103000981
|
[9] |
B. Bayour and D. F. M. Torres, "Existence of solution to a local fractional nonlinear differential equation, " J. Comput. Appl. Math. , vol. 312, pp. 127--133, Mar. 2017.
|
[10] |
N. Benkhettou, S. Hassani, and D. F. M. Torres, "A conformable fractional calculus on arbitrary time scales, " J. King Saud Univ. Sci. , vol. 28, no. 1, pp. 93-98, Jan. 2016. http://www.sciencedirect.com/science/article/pii/S1018364715000464
|
[11] |
F. Riewe, "Mechanics with fractional derivatives, " Phys. Rev. E, vol. 55, no. 3, pp. 3581-3592, Mar. 1997. http://adsabs.harvard.edu/abs/1997PhRvE..55.3581R
|
[12] |
P. S. Bauer, "Dissipative dynamical systems. I, " Proc. Natl. Acad. Sci. USA, vol. 17, pp. 311-314, Jun. 1931. doi: 10.1007%2FBF00276493
|
[13] |
O. P. Agrawal, "Formulation of Euler-Lagrange equations for fractional variational problems, " J. Math. Anal. Appl. , vol. 272, no. 1, pp. 368-379, Aug. 2002. http://www.sciencedirect.com/science/article/pii/S0022247X02001804
|
[14] |
R. Almeida and D. F. M. Torres, "Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, " Commun. Nonlinear Sci. Numer. Simul. , vol. 16, no. 3, pp. 1490-1500, 2011. http://philosophy.wisc.edu/hausman/341/Skill/nec-suf.htm
|
[15] |
D. Baleanu and O. P. Agrawal, "Fractional Hamilton formalism within Caputo's derivative, " Czechoslovak J. Phys. , vol. 56, no. 10-11, pp. 1087-1092, Oct. 2006. doi: 10.1007/s10582-006-0406-x
|
[16] |
J. Cresson, "Fractional embedding of differential operators and Lagrangian systems, " J. Math. Phys. , vol. 48, no. 3, pp. 033504, Mar. 2007. doi: 10.1063/1.2483292
|
[17] |
M. J. Lazo and D. F. M. Torres, "The DuBois-Reymond fundamental lemma of the fractional calculus of variations and an Euler-Lagrange equation involving only derivatives of Caputo, " J. Optim. Theory Appl. , vol. 156, no. 1, pp. 56-67, Jan. 2013. doi: 10.1007/s10957-012-0203-6
|
[18] |
T. Odzijewicz, A. B. Malinowska, and D. F. M. Torres, "Fractional variational calculus with classical and combined Caputo derivatives, " Nonlinear Anal. , vol. 75, no. 3, pp. 1507-1515, Feb. 2012. http://www.sciencedirect.com/science/article/pii/S0362546X11000113
|
[19] |
T. Odzijewicz, A. B. Malinowska, and D. F. M. Torres, "Fractional calculus of variations in terms of a generalized fractional integral with applications to physics, " Abstr. Appl. Anal. , vol. 2012, Article ID 871912, 2012. https://www.hindawi.com/journals/aaa/2012/871912/
|
[20] |
R. Almeida, S. Pooseh, and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations. London: Imperial College Press, 2015.
|
[21] |
A. B. Malinowska, T. Odzijewicz, and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations. New York: Springer International Publishing, 2015.
|
[22] |
A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations. London: Imperial College Press, 2012.
|
[23] |
M. J. Lazo and C. E. Krumreich, "The action principle for dissipative systems, " J. Math. Phys. , vol. 55, no. 12, pp. 122902, 2014. doi: 10.1063/1.4903991
|
[24] |
J. D. Logan, Invariant Variational Principles. Vol.138. Mathematics in Science and Engineering. New York, San Francisco, Lindon: Academic Press, 1977.
|
[25] |
D. F. M. Torres, "Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations, " Commun. Pure Appl. Anal. , vol. 3, no. 3, pp. 491-500, Sep. 2004. https://www.researchgate.net/publication/280005301_Proper_extensions_of_Noether%27s_symmetry_theorem_for_nonsmooth_extremals_of_the_calculus_of_variations
|
[26] |
G. S. F. Frederico and D. F. M. Torres, "Non-conservative Noether's theorem for fractional action-like variational problems with intrinsic and observer times, " Int. J. Ecol. Econ. Stat. , vol. 9, no. F07, pp. 74-82, Nov. 2007. http://www.academia.edu/2372009/Non-conservative_Noethers_theorem_for_fractional_action-like_variational_problems_with_intrinsic_and_observer_times
|
[27] |
G. S. F. Frederico and D. F. M. Torres, "A formulation of Noether's theorem for fractional problems of the calculus of variations, " J. Math. Anal. Appl. , vol. 334, no. 2, pp. 834-846, Oct. 2007. http://www.sciencedirect.com/science/article/pii/S0022247X07000340
|
[28] |
G. S. F. Frederico and D. F. M. Torres, "Fractional conservation laws in optimal control theory, " Nonlinear Dyn. , vol. 53, no. 3, pp. 215-222, Aug. 2008. doi: 10.1007/s11071-007-9309-z
|
[29] |
G. S. F. Frederico and D. F. M. Torres, "Fractional optimal control in the sense of Caputo and the fractional Noether's theorem, " Int. Math. Forum, vol. 3, no. 10, pp. 479-493, Sep. 2008. http://www.wenkuxiazai.com/doc/e6c1f7fec8d376eeaeaa31f2-2.html
|
[30] |
G. S. F. Frederico and D. F. M. Torres, "Fractional Noether's theorem in the Riesz-Caputo sense, " Appl. Math. Comput. , vol. 217, no. 3, pp. 1023-1033, Oct. 2010. http://www.sciencedirect.com/science/article/pii/S0096300310001244
|
[31] |
N. Benkhettou, A. M. C. Brito Da Cruz, and D. F. M. Torres, "A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration, " Signal Proc. , vol. 107, pp. 230-237, Feb. 2015. http://www.sciencedirect.com/science/article/pii/S0165168414002448
|
[32] |
N. Benkhettou, A. M. C. Brito da Cruz, and D. F. M. Torres, "Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets, " Math. Methods Appl. Sci. , vol. 39, no. 2, pp. 261-279, Jan. 2016. http://www.oalib.com/paper/3941045
|
[33] |
W. Sarlet and F. Cantrijn, "Generalizations of Noether's theorem in classical mechanics, " SIAM Rev. , vol. 23, no. 4, pp. 467-494, 1981. doi: 10.1137/1023098
|
[34] |
G. S. F. Frederico and D. F. M. Torres, "Nonconservative Noether's theorem in optimal control, " Int. J. Tomogr. Stat. , vol. 5, no. W07, pp. 109-114, 2007. http://www.academia.edu/2372000/Nonconservative_Noethers_theorem_in_optimal_control
|
[35] |
D. F. M. Torres, "On the Noether theorem for optimal control, " Eur. J. Control, vol. 8, no. 1, pp. 56-63, 2002.
|
[36] |
D. F. M. Torres, "Conservation laws in optimal control, " in Dynamics, Bifurcations, and Control, vol. 273, F. Colonius and L Grüne, Eds. Berlin: Springer, 2002, pp. 287-296.
|
[37] |
D. F. M. Torres, "Quasi-invariant optimal control problems, " Port. Math. , vol. 61, no. 1, pp. 97-114, 2004. https://archive.org/details/arxiv-math0302264
|
[38] |
S. Pooseh, R. Almeida, and D. F. M. Torres, "Fractional order optimal control problems with free terminal time, " J. Ind. Manag. Optim. , vol. 10, no. 2, pp. 363-381, Apr. 2014. doi: 10.1186/s13662-016-0976-2
|
[39] |
D. S. Dukić, "Noether's theorem for optimum control systems, " Int. J. Control, vol. 18, no. 3, pp. 667-672, Jul. 1973. doi: 10.1080/00207177308932544
|
[40] |
A. S. Balankin, J. Bory-Reyes, and M. Shapiro, "Towards a physics on fractals: differential vector calculus in three-dimensional continuum with fractal metric, " Phys. A, vol. 444, pp. 345-359, Feb. 2016. http://www.sciencedirect.com/science/article/pii/S037843711500881X
|
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[11] | Bingsan Chen, Chunyu Li, Benjamin Wilson, Yijian Huang. Fractional Modeling and Analysis of Coupled MR Damping System[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 288-294. |
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[13] | YangQuan Chen, Dingyü Xue, Antonio Visioli. Guest Editorial for Special Issue on Fractional Order Systems and Controls[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 255-256. |
[14] | Jiacai Huang, YangQuan Chen, Haibin Li, Xinxin Shi. Fractional Order Modeling of Human Operator Behavior with Second Order Controlled Plant and Experiment Research[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 271-280. |
[15] | Norelys Aguila-Camacho, Manuel A. Duarte-Mermoud. Improving the Control Energy in Model Reference Adaptive Controllers Using Fractional Adaptive Laws[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 332-337. |
[16] | Kai Chen, Junguo Lu, Chuang Li. The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 311-319. |
[17] | Bruce J. West, Malgorzata Turalska. The Fractional Landau Model[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 257-260. |
[18] | Mojtaba Naderi Soorki, Mohammad Saleh Tavazoei. Constrained Swarm Stabilization of Fractional Order Linear Time Invariant Swarm Systems[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 320-331. |
[19] | Cuihong Wang, Huanhuan Li, YangQuan Chen. H∞ Output Feedback Control of Linear Time-invariant Fractional-order Systems over Finite Frequency Range[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(3): 304-310. |
[20] | Fudong Ge, YangQuan Chen, Chunhai Kou. Cyber-physical Systems as General Distributed Parameter Systems: Three Types of Fractional Order Models and Emerging Research Opportunities[J]. IEEE/CAA Journal of Automatica Sinica, 2015, 2(4): 353-357. |