Fractional Herglotz variational problems of variable order

We study fractional variational problems of Herglotz type of variable order. Necessary optimality conditions, described by fractional differential equations depending on a combined Caputo fractional derivative of variable order, are proved. Two different cases are considered: the fundamental problem, with one independent variable, and the general case, with several independent variables. We end with some illustrative examples of the results of the paper.


1.
Introduction. The theory of fractional calculus is an extension of ordinary calculus that considers integrals and derivatives of arbitrary real or complex order. Although its birth goes back to Euler, fractional calculus has gained a great importance only in recent decades, with the applicability of such operators for the efficient dynamic modeling of some real phenomena [2,18]. More recently, a general theory of fractional calculus was presented, where the order of the fractional operators is not constant in time [12]. This is a natural extension, since fractional derivatives are nonlocal operators and contain memory. Therefore, it is reasonable that the order of the derivative may vary along time.
The variational problem of Herglotz is a generalization of the classical variational problem [16,17]. It allows us to describe nonconservative processes, even in the case when the Lagrangian is autonomous (that is, when the Lagrangian does not depend explicitly on time). In contrast to the calculus of variations, where the cost functional is given by an integral depending only on time, space and velocity, in the Herglotz variational problem the model is given by a differential equation involving the derivative of the objective functional z and the Lagrange function depends on time, trajectories x and z and on the derivative of x. The problem of Herglotz was posed by Herglotz himself in 1930 [7], but only in 1996, with the works [5,6], it has gained a wide attention from the mathematical community. Indeed, since 1996, several papers were devoted to this subject: see [1,3,4,13,14,15,16,17] and references therein.
2. Preliminaries. In this section we present some needed concepts and results.
2.1. The fractional calculus of variable order. We deal with fractional operators of variable fractional order on two variables, with range on the open interval (0, 1), that is, the order is a function α : [a, b] 2 → (0, 1). Given a function x : [a, b] → R, we present two different concepts of fractional derivatives of x. First, we recall the definition of fractional integral [10].
For fractional derivatives, we consider two types: Riemann-Liouville and Caputo fractional derivatives.
and the right Riemann-Liouville fractional derivative of x by Definition 2.3. The left Caputo fractional derivative of order α(·, ·) of x is defined by Motivated by the works [8,9], we consider here a generalization of previous concepts by introducing a linear combination of the fractional derivatives of variable fractional order.
(1) Similarly, the combined Caputo fractional derivative of function x is defined by When dealing with variational problems and necessary optimality conditions, an important ingredient is always an integration by parts formula. Here, we present two such formulas, involving the Caputo fractional derivative of variable order.
To end this short introduction to the fractional calculus of variable order, we introduce one more notation. The dual fractional derivative of (1) is defined by where γ = (γ 2 , γ 1 ) and T ∈ [a, b] is the final time of the problem under consideration (see (3) below). The dual fractional derivative of (2) is defined similarly.
To fix notation, throughout the text we denote by ∂ i ψ the partial derivative of a function ψ : R n → R with respect to its ith argument, i = 1, . . . , n. For simplicity of notation, we also introduce the operator [·] α,β γ defined by Let L be a Lagrangian L : Consider the following problem of the calculus of variations: minimize functional J : over all (x, T ) ∈ D satisfying the initial condition x(a) = x a , for a given x a ∈ R. The terminal time T and the terminal state x(T ) are considered here free. The Theorem 2.6 (See [19]). If (x, T ) is a minimizer of functional (3) on D, then (x, T ) satisfies the fractional differential equations on [a, T ] and Moreover, the following transversality conditions hold: We can rewrite the transversality conditions (6), obtaining the next result.
then the fractional Euler-Lagrange equations (4) and (5) are satisfied together with the following transversality conditions: 3. Herglotz's variational principle. In this section we present a fractional variational principle of Herglotz depending on Caputo fractional derivatives. Let α, β : [a, b] 2 → (0, 1) be two functions. The fractional Herglotz variational problem that we study consists in the determination of trajectories x ∈ C 1 ([a, b]), satisfying a given initial condition x(a) = x a ∈ R, and a real T ∈ [a, b] that extremize the value of z(T ), where z satisfies the following differential equation with dependence on a combined Caputo fractional derivative operator: subject to the initial condition z(a) = z a , (8) where z a is a given real number. In the sequel, we use the auxiliary notation The Lagrangian L is assumed to satisfy the following hypothesis: The following result gives necessary conditions of Euler-Lagrange type for an admissible function x to be solution of the problem.
) be such that z defined by (7) subject to the initial condition (8) has an extremum. Then, (x, z) satisfies the fractional differential equations Moreover, the following transversality conditions are satisfied: Proof. Let x be a solution to the problem and consider an admissible variation of is an arbitrary perturbation curve and ǫ ∈ R represents a small number (ǫ → 0). The constraint x(a) = x a implies that all admissible variations must fulfill the condition h(a) = 0. On the other hand, consider an admissible variation of z, z = z + ǫθ, where θ is a perturbation curve (not arbitrary) such that , so that the variation satisfies equation (7).
Differentiating θ with respect to t, we obtain that and rewriting this relation, we obtain the following differential equation for θ: Considering we obtain the solution for the last differential equation: By hypothesis, θ(a) = 0. If x is such that z(x, t) defined by (7) attains an extremum at t = T , then θ(T ) is identically zero. Hence, we get Considering only the second term in (12), and the definition of combined Caputo derivative, we obtain that Using Theorem 2.5, and considering γ = (γ 2 , γ 1 ), we get Substituting this relation into expression (12), we obtain T a h(t) ∂ 2 L[x, z] α,β γ (t)λ(t) + D β(·,·),α(·,·) γ With appropriate choices for the variations h(·), we get the Euler-Lagrange equations (9)-(10) and the transversality conditions (11).

Remark 1.
If α(·, ·) and β(·, ·) tend to 1, and if the Lagrangian L is of class C 2 , then the first Euler-Lagrange equation (9) becomes Differentiating and considering the derivative of the lambda function, we get As λ(t) > 0 for all t, we deduce that 4. The case of several independent variables. We now obtain a generalization of Herglotz's principle of Section 3 for problems involving n + 1 independent vari- × Ω and consider the vector s = (s 1 , s 2 , . . . , s n ) ∈ Ω. The new problem consists in determining the trajectories x ∈ C 1 (P ) that give an extremum to z[x, T ], where the functional z satisfies the differential equatioṅ x(t, s), . . . , C D αn(·,·),βn(·,·) γ n x(t, s), z(t) d n s (13) subject to the constraint where ∂P is the boundary of P and g is a given function g : x(t, s), . . . , C D αn(·,·),βn(·,·) γ n x(t, s) exist and are continuous functions, 5. the Lagrangian L : P × R n+3 → R is of class C 1 .
x(t, s) we mean the Caputo fractional derivative with respect to the independent variable t and by C D αi(·,·),βi(·,·) γ i x(t, s) we mean the Caputo fractional derivative with respect to the independent variable s i , i = 1, . . . , n.
Theorem 4.1. If (x, z) is an extremizer to the functional (13), then (x, z) satisfies the fractional differential equation Proof. Let x be a solution to the problem. Consider an admissible variation of x, x(t, s) = x(t, s) + ǫh(t, s), where h ∈ C 1 (P ) is an arbitrary perturbing curve and ǫ ∈ R is such that |ǫ| ≪ 1. Consequently, from the boundary condition (14), h(t, s) = 0 for all (t, s) ∈ ∂P . On the other hand, consider an admissible variation of z, z = z + ǫθ, where θ is a perturbing curve such that θ(a) = 0 and .
To simplify the notation, define Then, we obtain the linear differential equatioṅ whose solution is Since θ(a) = θ(T ) = 0, we get Considering only the second term in (18), we can write h(t, s)d n sdt .
In this case, λ(t) ≡ 1. We intend to find a pair (x, z), satisfying all the conditions in (22), for which z(T ) attains a minimum value. It is easy to verify that x(t) = t and T = 1 satisfy the necessary conditions given by Theorem 3.1. Replacing x by x in system (22), we get a Cauchy problem of forṁ z(0) = 0, whose solution is Observe that this function attains a minimum value at T = 1, which is z(1) = −2/3 ( Figure 1, (c)).