A Survey on Fractional Variational Calculus 3

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2 The fundamental problem

The calculus of variations deals with functional optimization problems involving an unknown function x and its derivative x′. The general way of formulating the problem is as follows:

b min J(x)= L(t, x(t), x′(t)) dt Z a subject to the boundary conditions

x(a)= xa and x(b)= xb, with xa, xb ∈ R. (1)

One way to solve this problem consists to determine the solutions of the second order diﬀerential equation d ∂ L(t, x(t), x′(t)) = ∂ L(t, x(t), x′(t)), t ∈ [a,b], (2) 2 dt 3 together with the boundary conditions (1). Equation (2) was studied by Euler, and later by Lagrange, and is known as the Euler–Lagrange equation. A survey on fractional variational calculus 3

The fractional calculus of variations is a generalization of the ordinary variational calculus, where the integer-order derivative is replaced by a fractional derivative Dαx: b min J(x)= L(t, x(t),Dαx(t)) dt. Z a As Riewe noted in [25], “traditional Lagrangian and Hamiltonian mechanics cannot be used with nonconservative forces such as friction”. FVC allows it in an elegant way. Since there are several definitions of fractional derivatives, one finds several works on the calculus of variations for these different operators. Here we consider the Caputo fractional derivative [11], but analogous results can be formulated for other types of fractional derivatives. Our choice of derivative is based on the following important fact: the Laplace transform of the Caputo fractional derivative depends on integer-order derivatives evaluated at the initial time t = a. Thus, we have an obvious physical interpretation for them. For other types of differentiation, this may not be so clear. For example, when dealing with the Riemann–Liouville fractional derivative, the initial conditions are fractional. The main problem of the fractional calculus of variations, in the context of the Caputo fractional derivative, is stated as follows. Given a functional

b J(x)= L t, x(t), CDα x(t) dt, x ∈ Ω, (3) Z a+ a where Ω is a given class of functions, ﬁnd a curve x for which J attains a minimum value. We suppose that L : [a,b] × R2 → R is continuously diﬀerentiable with respect to the second and third variables and, for every x ∈ Ω, the function C α Da+x exists and is continuous on the interval [a,b]. On the set of admissible functions of the problem, the boundary conditions

x(a)= xa and x(b)= xb, with xa, xb ∈ R, (4) may be imposed. We observe that a curve x⋆ ∈ Ω is a (local) minimizer of functional (3) if there exists some positive real δ such that, whenever x ∈ Ω and kx⋆ − xk < δ, one has J(x⋆) − J(x) ≤ 0.

3 Indirect methods

The literature about the fractional calculus of variations is very vast [4, 5, 18, 19]. The usual procedure in the calculus of variations, to find a candidate to 4 R. Almeida and D. F. M. Torres minimizer, consists to take variations of the optimal curve x⋆. Let ǫ be a real number close to zero and η a function chosen so that x⋆ + ǫη belongs to Ω. For example, if the functional is defined in a class of functions with the boundary conditions (4), then the curve η must satisfy the conditions η(a) =0and η(b)=0. The curve x⋆ +ǫη is called a variation of the curve x⋆. For simplicity of notation, we will use the operator [ · ]α defined by

α C α [x] (t) := t, x(t), Da+x(t) . Suppose that x⋆ minimizes J, and consider a variation x⋆ + ǫη. Depending on the existence of boundary conditions, some constraints over η(a) and η(b) may be imposed. Consider a function j, deﬁned on a neighborhood of zero, and given by

b j(ǫ) := J(x⋆ + ǫη)= L(t, x⋆(t)+ ǫη(t), CDα x⋆(t)+ ǫC Dα η(t)) dt. Z a+ a+ a Since J attains a minimum value at x⋆, then we conclude that ǫ = 0 is a minimizer of j, and thus j′(0) = 0. Since

b ′ ⋆ α ⋆ α C α j (ǫ)= ∂2L[x + ǫη] (t)η(t)+ ∂3L[x + ǫη] (t) D η(t) dt, Z a+ a we conclude that b ⋆ α ⋆ α C α ∂2L[x ] (t)η(t)+ ∂3L[x ] (t) D η(t) dt =0. (5) Z a+ a Using now fractional integration by parts [16], we obtain that

b ⋆ α C α ∂3L[x ] (t) D η(t) dt Z a+ a b α ⋆ α 1−α ⋆ α b = D (∂3L[x ] (t)) η(t) dt + I (∂3L[x ] (t)) η(t) . (6) Z b− b− a a Combining equations (5) and (6), we prove that

b ⋆ α α ⋆ α α ∂2L[x ] (t)+ D (∂3L[x ] (t)) η(t) dt Z b− a b + I1−α (∂ L[x⋆]α(t)) η(t) =0. (7) b− 3 a A survey on fractional variational calculus 5

Assume that the set of admissible functions have to satisfy the boundary constraints (4). In this case, the variation function η must satisfy the conditions η(a)=0 and η(b) = 0. Thus, equation (7) becomes

b ⋆ α α ⋆ α ∂2L[x ] (t)+ D (∂3L[x ] (t)) η(t) dt =0. Z b− a

α ⋆ α Assume now that the function t 7→ Db− (∂3L[x ] (t)) is continuous on [a,b]. Since η is an arbitrary function on the open interval ]a,b[, we conclude, by the fundamental lemma of the calculus of variations [14], that

⋆ α α ⋆ α ∂2L[x ] (t)+ Db− (∂3L[x ] (t))=0 (8) for all t ∈ [a,b]. Assume now that x(a) and x(b) are free. By the arbitrariness of η, we can ﬁrst consider variations such that η(a)=0 and η(b) = 0, and in this case we prove equation (8). Therefore, replacing (8) in (7), we deduce that

b I1−α (∂ L[x⋆]α(t)) η(t) dt =0. b− 3 a If η(a)=0 and η(b) 6= 0, then

1−α ⋆ α Ib− (∂3L[x ] (t))=0 at t = b, and if η(a) 6= 0 and η(b) = 0, then

1−α ⋆ α Ib− (∂3L[x ] (t))=0 at t = a.

In conclusion, we proved the following result, which provides a necessary optimality condition, known as the fractional Euler–Lagrange equation, and two subsidiary conditions, known as transversality conditions.

Theorem 1. Let x⋆ be an admissible function and suppose that J attains a minimum value at x⋆. If there exists and is continuous the function t 7→ α ⋆ α Db− (∂3L[x ] (t)), then

⋆ α α ⋆ α ∂2L[x ] (t)+ Db− (∂3L[x ] (t))=0 (9) for every t ∈ [a,b]. Moreover, if x(a) is free, then

1−α ⋆ α Ib− (∂3L[x ] (t))=0 at t = a and, if x(b) is free, then

1−α ⋆ α Ib− (∂3L[x ] (t))=0 at t = b. 6 R. Almeida and D. F. M. Torres

As observed in [7], using the well–know relation

f(b) Dα f(t)= C Dα f(t)+ Dα f(b)= C Dα f(t)+ , b− b− b− b− Γ(1 − α)(b − t)α between Riemann–Liouville and Caputo derivatives, one can easily rewrite the the Euler–Lagrange equation (9) in the equivalent form

∂ L[x⋆]α(t)| ∂ L[x⋆]α(t)+ C Dα (∂ L[x⋆]α(t)) + 3 t=b =0. 2 b− 3 Γ(1 − α)(b − t)α Following similar ideas, we can study the case for several dependent variables (x1(t),...,xm(t)), with m ∈ N. In this case, we obtain m fractional diﬀerential equations, one for each function xi, i =1,...,m.

Theorem 2. Let J be the functional

b α J(x1,...,xm) := L[x] (t) dt Z m a with α C α1 C αm [x]m(t) := t, x1(t),...,xm(t), Da+x1(t),..., Da+ xm(t) , where α1,...,αm ∈ ]0, 1[ and m ∈ N, deﬁned on the set of functions

1 m x ∈ C ([a,b], R ) : x(a)= xa ∧ x(b)= xb , m 2m where xa, xb are two ﬁxed vectors in R . Suppose also that L :[a,b] × R → R is continuously diﬀerentiable with respect to its ith variable, i = 2,..., 2m + ⋆ ⋆ 1. Let (x1,...,xm) be a minimizer of J, and suppose that the functions t 7→ αi ⋆ α Db− (∂i+1+mL[x ]m(t)) exist and are continuous on [a,b], i =1,...,m. Then,

⋆ α αi ⋆ α ∂i+1L[x ]m(t)+ Db− (∂i+1+mL[x ]m(t))=0 for all i =1,...,m and for all t ∈ [a,b].

In the next result, the lower bound of the cost functional is a real A, where A>a, that is, A is greater than the lower bound of the fractional derivative. For simplicity of presentation, we restrict ourselves to the case m = 1.

Theorem 3. Deﬁne the functional A as

b J(x) := L[x]α(t) dt, Z A A survey on fractional variational calculus 7 where A> a. If J achieves a minimum value at x⋆, and if the maps t 7→ α ⋆ α α ⋆ α DA− (∂3L[x ] (t)), for t ∈ [a, A], and t 7→ Db− (∂3L[x ] (t)), for [a,b], exist and are continuous, then

α ⋆ α α ⋆ α Db− (∂3L[x ] (t)) − DA− (∂3L[x ] (t))=0, t ∈ [a, A], and ⋆ α α ⋆ α ∂2L[x ] (t)+ Db− (∂3L[x ] (t))=0, t ∈ [A, b]. Moreover, the following transversality conditions are fulfilled: 1−α ⋆ α 1−α ⋆ α Ib− (∂3L[x ] (t)) − IA− (∂3L[x ] (t))=0 at t = a, if x(a) is free; 1−α ⋆ α  IA− (∂3L[x ] (t))=0 at t = A, if x(A) is free;  1−α ⋆ α Ib− (∂3L[x ] (t))=0 at t = b, if x(b) is free.  For our next result, besides the boundary conditions, an integral constraint is imposed on the set of admissible functions. Such type of problems are known as isoperimetric problems, and the first example goes back to Dido, Queen of Cartage in Africa. She was interested if finding the shape of a curve with fixed perimeter, of maximum possible area. As we know, the solution is given by a circle. The calculus of variations presents a solution to such problem, with an integral constraint of type b 1+(x′(t))2 dt = constant. Z a p In our case, we consider that the new constraint depends also on a fractional derivative, and it is of form b G(x) := M t, x(t), CDα x(t) dt = K, K ∈ R, (10) Z a+ a where M :[a,b]×R2 → R is a constant, such that there exist and are continuous the functions ∂2M and ∂3M.

Theorem 4. Suppose that J (3), subject to the constraints (4) and (10), attains a minimum value at x⋆. If x⋆ is not a solution for

α α α ∂2M[x] (t)+ Db− (∂3M[x] (t))=0, ∀t ∈ [a,b], (11) α ⋆ α and if there exist and are continuous the functions t 7→ Db− (∂3L[x ] (t)) and α ⋆ α R ⋆ t 7→ Db− (∂3M[x ] (t)) on [a,b], then there exists λ ∈ such that x satisﬁes α α α ∂2F [x] (t)+ Db− (∂3F [x] (t))=0, ∀t ∈ [a,b], where we deﬁne the function F as F := L + λM. 8 R. Almeida and D. F. M. Torres

The case when x⋆ is a solution of (11) can be easily included:

Theorem 5. Suppose that J (3), subject to the constraints (4) and (10), attains a minimum value at x⋆. If there exist and are continuous the functions t 7→ α ⋆ α α ⋆ α Db− (∂3L[x ] (t)) and t 7→ Db− (∂3M[x ] (t)) on [a,b], then there exist λ0, λ ∈ R, not both zero, such that x⋆ satisﬁes the equation

α α α ∂2F [x] (t)+ Db− (∂3F [x] (t))=0, ∀t ∈ [a,b], where we deﬁne the function F as F := λ0L + λM.

Next, we present another constrained type problem, but now the restriction is given by g(t, x(t))=0, ∀t ∈ [a,b], (12)

2 where x = (x1, x2) is a vector and g : [a,b] × R → R is diﬀerentiable with respect to x1 and x2. Also, we have the boundary conditions

2 x(a)= xa and x(b)= xb, xa, xb ∈ R . (13)

Theorem 6. Consider the functional

b J(x)= L[x]α(t) dt, Z 2 a where α C α C α [x]2 (t) := t, x1(t), x2(t), Da+x1(t), Da+x2(t) , deﬁned on C1[a,b]×C1[a,b], subject to the constraints (12) and (13). If J attains ⋆ ⋆ ⋆ α ⋆ α an extremum at x =(x1, x2), the maps t 7→ Db− (∂i+3L[x ]2 (t)), i =1, 2, are continuous, and ∂3g(t, x(t)) 6=0 for all t ∈ [a,b], then there exists a continuous function λ :[a,b] → R such that

⋆ α C α ⋆ α ∂i+1L[x ]2 (t)+ Db− (∂i+3L[x ]2 (t)) + λ∂i+1g(t, x(t))=0 for all t ∈ [a,b] and i =1, 2.

Infinite horizon problems are an important field of research, that deal with phenomena that spread along time [1, 13, 20]. In this case, the cost functional is evaluated in an infinite interval [a, ∞[:

∞ J(x) := L[x]α dt, (14) Z a A survey on fractional variational calculus 9

deﬁned on a set of functions with ﬁxed initial conditions: x(a) = xa. Since we are dealing with improper integrals, some attention to what is a minimizer is needed. We say that x⋆ is a local minimizer for J as in (14) if there exists some ǫ> 0 such that, for all x, if kx⋆ − xk < ǫ, then

b lim inf [L[x⋆]α(t) − L[x]α(t)] dt ≤ 0. T →∞ b≥T Z a Also, let us deﬁne the functions

b L[x⋆ + ǫv]α(t) − L[x⋆]α(t) A(ǫ, b) := dt; Z ǫ a b V (ǫ, T ) := inf [L[x⋆ + ǫv]α(t) − L[x⋆]α(t)] dt; b≥T Z a W (ǫ):= lim V (ǫ, T ), T →∞ where v ∈ C1[a, ∞[ is a function and ǫ a real.

Theorem 7. Let x⋆ be a local minimal for J as in (14). Suppose that: V (ǫ, T ) 1. lim exists for all T ; ǫ→0 ǫ V (ǫ, T ) 2. lim exists uniformly for all ǫ; T →∞ ǫ 3. For every T >a and ǫ 6=0, there exists a sequence (A(ǫ, bn))n∈N such that

lim A(ǫ, bn)= inf A(ǫ, b) n→∞ b≥T

uniformly for ǫ. α ⋆ α If there exists and is continuous the function t 7→ Db− (∂3L[x ] (t)) on [a,b], for all b>a, then

⋆ α α ⋆ α ∂2L[x ] (t)+ Db− (∂3L[x ] (t))=0, for all b>a. Also, we have

1−α ⋆ α lim inf I − (∂3L[x ] (t))=0 at t = b. T →∞ b≥T b

So far, we considered variational problems with order α ∈]0, 1[. Now we proceed by extending them to functionals depending on higher-order derivatives. To that 10 R. Almeida and D. F. M. Torres purpose, consider the functional b J(x) := L t, x(t), CDα1 x(t),..., C Dαm x(t) dt, (15) Z a+ a+ a defined on Cm[a,b], such that x(i)(a) and x(i)(b) are fixed reals for i ∈ {0, 1,...,m−1}. Here, m is a positive integer, αi ∈ (i−1, i), for all i ∈{1,...,m}, and L :[a,b] × Rm+1 → R is differentiable with respect to the ith variable, for i ∈{2, 3,...,m +1}.

Theorem 8. Let x⋆ be a minimizer of J (15), and suppose that for all i ∈ αi ⋆ α {1,...,m}, there exist and are continuous the functions t 7→ Db− (∂i+2L[x ]m(t)) on [a,b]. Then, m ⋆ α αi ⋆ α ∂2L[x ]m(t)+ Db− (∂i+2L[x ]m(t))=0 Xi=1 ⋆ α C α1 ⋆ C αm ⋆ for all t ∈ [a,b], where [x ]m(t) := t, x(t), Da+x (t),..., Da+ x (t) . After solving the presented Euler–Lagrange equations, we still need to verify if we are in presence of a minimizer of the functional or not. We recall that the Euler–Lagrange equation is just a necessary optimality condition of ﬁrst order, and thus its solutions may not be a solution for the problem. For the question of existence of solutions, we refer to [8, 9]. One possible way to check if we have a candidate for minimizer or maximizer is to apply the Legendre condition, which is a second order necessary optimality condition [3, 17]. Under suitable convexity of the Lagrangian, suﬃcient conditions for global minimizer hold [6]. For the Legendre condition, we have to assume that the Lagrange function 2 L is such that its second order partial derivatives ∂ijL, with i, j ∈ {2, 3}, exist and are continuous.

Theorem 9. Suppose that x⋆ is a local minimum for J as in (3). Then, 2 ⋆ α ∂33L[x ] (t) ≥ 0, ∀t ∈ [a,b].

For our next result, we recall that a function L :[a,b] × R2 → R is convex with respect to the second and third variables if

L(t, x + v,y + w) − L(t,x,y) ≥ ∂2L(t,x,y)v + ∂3L(t,x,y)w for all t ∈ [a,b] and x,y,v,w ∈ R.

Theorem 10. If L is convex and if x⋆ satisﬁes the Euler–Lagrange equation (9), then x⋆ minimizes J (3) when restricted to the boundary conditions (4). A survey on fractional variational calculus 11

4 Direct methods

There are several different numerical approaches and methods to solve fractional differential equations [2, 10, 12]. Different discretizations of fractional derivatives are possible, but many of them do not preserve the fundamental properties of the systems, such as stability [15, 29]. For numerical calculations, a simple and powerful method preserving stability is obtained from Grünwald–Letnikov discretizations. For a detailed numerical treatment of fractional differential equations, based on Grünwald–Letnikov fractional derivatives, we refer the interested reader to [24, 28]. Here we just mention that both explicit and implicit methods are possible: Theorem 5.1 of [28] shows that the explicit and implicit Grünwald– Letnikov methods are asymptotically stable; while Theorem 5.2 of [28] gives conditions on the step size for the explicit method to be absolute stable, assert- ing that the implicit method is always absolute stable, without any step size restriction. It is also worth to underline the good convergence properties and error behavior of Grünwald–Letnikov methods [28]. We start this section by recalling the (left) Grünwald–Letnikov fractional derivative of a function x. Let α> 0 be a real. The Grünwald–Letnikov fractional derivative of order α is defined by ∞ GL α 1 k α a Dx x(t):= lim (−1) x(t − kh), h→0+ hα k kX=0 α where k stands for the generalization of binomial coefficients to real numbers. As usual, we will adopt the notation α (wα):=(−1)k . k k This fractional derivative is particularly useful to approximate the Caputo derivative. The method is now briefly explained. Given an interval [a,b] and a fixed integer N, let tj := a+jh, j =0, 1,...,N and h> 0, be a partition of the interval [a,b]. Then,

j C α 1 α x(a) −α D x(t )= (w )x(t − ) − (t − a) + O(h). a+ j hα k j k Γ(1 − α) j kX=0 Thus, truncated Grünwald–Letnikov fractional derivatives are ﬁrst-order approximations of the Caputo fractional derivatives. The approximation used for the Caputo derivative is then j C α 1 α x(a) −α D x(t ) ≈ (w )x(t − ) − (t − a) := Dx˜ (t ). a+ j hα k j k Γ(1 − α) j j Xk=0 12 R. Almeida and D. F. M. Torres

The next step is to discretize the functional (3). For simplicity, let h =(b−a)/N and let us consider the grid tj = a + jh, j =0, 1,...,N. Then,

N tk J(x) = L(t, x(t), CDα x(t)) dt Z a+ k=1 Xtk−1 N C α (16) ≈ hL(tk, x(tk), Da+x(tk)) kX=1 N ≈ hL(tk, x(tk), Dx˜ (tk)) dt. kX=1 The right hand side of (16) can be regarded as a function Ψ of N − 1 unknowns:

N Ψ(x1, x2,...,xN−1)= hL(tk, x(tk), Dx˜ (tk)). Xk=1 To ﬁnd an extremum for Ψ, one has to solve the following system of algebraic equations: ∂Ψ =0, i =1,...,N − 1. ∂xi Suppose that h goes to zero. The solution obtained by this method converges to a function x⋆. Then, x⋆ is a solution of the Euler–Lagrange equation (9) (for details, see [4, Theorem 8.1]).

Example 1. Consider the functional

10 2 2 J(x)= C D0.5x(t) − t1.5 dt Z 0+ Γ(3/2) 0

C 0.5 2 subject to the boundary conditions x(0) = 0 and x(10) = 100. Since D0+ t = 2 1.5 J 2 Γ(3/2) t , and is nonnegative, we conclude that the function x(t) = t is a minimizer of the functional. If we discretize the functional, for diﬀerent values of n, we obtain several numerical approximations of x. In Table 1 we present the error of each n, where the error is given by the maximum of the absolute value of the diﬀerence between x and the numerical approximation.

Example 2. For our second example, we consider a problem where we do not know its exact solution. Let

1 2 C 0.5 2 J(x)= x(t) D0+ x(t) − sin(x(t)) dt Z 0 REFERENCES 13

Table 1: Errors for the numerical solution of problem of Example 1.

n 10 50 100 200 Error 0.425189 0.107622 0.056370 0.029215 subject to the boundary conditions x(0) = 0 and x(1) = 1. In Figure 1, we present the results for n = 100.

Fig. 1: Plot of the numerical solution of problem of Example 2.

Acknowledgment: the authors are grateful to the Research and Development Unit UID/MAT/04106/2013 (CIDMA) and to two anonymous referees for several constructive comments.

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