Generalized fractional isoperimetric problem of several variables

This work deals with the generalized fractional calculus of variations 
of several variables. Precisely, we prove a sufficient optimality condition for the 
fundamental problem and a necessary optimality condition for 
the isoperimetric problem. Our results cover important particular cases of problems 
with constant and variable order fractional operators.


1.
Introduction. Within the years, several methods were proposed to solve mechanical problems with nonconservative forces, e.g., Rayleigh dissipation function method, technique introducing an auxiliary coordinate or approach including the microscopic details of the dissipation directly in the Lagrangian. Although, all mentioned methods are correct, they are not as direct and simple as it is in the case of conservative systems. In the notes from 1996-1997, Riewe presented a new approach to nonconservative forces [22,23], where he claimed that friction forces follow from Lagrangians containing terms proportional to fractional derivatives. Riewe considered energy functionals containing fractional derivatives and with his works he initiated the mathematical field that is now called Fractional Calculus of Variations (FCV).
FCV unifies the calculus of variations and the fractional calculus by inserting fractional derivatives (and/or integrals) into the variational functionals. Fractional integrals and derivatives can be defined in different ways, and in each of the cases one needs to study different variational problems. By now several approaches were developed and the results include problems depending on e.g., Caputo fractional derivatives, Riemann-Liouville fractional derivatives, Riesz fractional derivatives, Hadamard fractional derivatives, variable order fractional derivatives [2-10, 14, 16, 17]. Therefore, in order to unify available results, in this work we study more general variational problems, where integral functionals depend on an unknown function (or functions) of several variables and its generalized partial fractional derivatives and/or generalized partial fractional integrals. Precisely, we are interested in the generalized fractional isoperimetric problem of several variables, where admissible functions need to satisfy certain boundary conditions and an isoperimetric constraint.
The work is organized as follows. In Section 2 we give basic definitions of ordinary and partial generalized fractional operators as well as generalized fractional integration by parts formulas. Section 3 is devoted to the generalized fractional variational problems of several variables. Precisely, we study fundamental and isoperimetric problems: proving a necessary (Theorem 3.4 and Theorem 3.6) and sufficient optimality conditions (Theorem 3.3 and Theorem 3.5). Except main theorems, several corollaries concerning standard fractional operators are given.

2.
Preliminaries. This section describes definitions of generalized fractional operators. In special cases, these operators simplify to the classical fractional integrals and derivatives. Interested reader can find more information in the following works [1,15,[19][20][21].
2.1. Generalized fractional operators. In order to define generalized fractional operators let us introduce the following triangle Definition 2.1. Let us consider a function k : ∆ → R. For any function f : (a, b) → R, the generalized fractional integral operator K P is defined for all t ∈ (a, b) by: with P = a, t, b, λ, µ , λ, µ ∈ R.
The generalized differential operators A P and B P are defined with the help of the operator K P .
Definition 2.2. The generalized fractional derivative of Riemann-Liouville type, denoted by A P , is defined by The next differential operator is obtained by interchanging the order of the operators in the composition that defines A P . Definition 2.3. The general kernel differential operator of Caputo type, denoted by B P , is given by Example 2. The standard Riemann-Liouville and Caputo fractional derivatives (see, e.g., [13,14]) are easily obtained from the general kernel operators A P and B P , respectively. Let k α (t − τ ) = 1 Γ(1−α) (t − τ ) −α , α ∈ (0, 1). If P = a, t, b, 1, 0 , then is the standard left Riemann-Liouville fractional derivative of order α, while is the standard left Caputo fractional derivative of order α; if P = a, t, b, 0, 1 , then is the standard right Riemann-Liouville fractional derivative of order α, while is the standard right Caputo fractional derivative of order α.

2.2.
Generalized partial fractional operators. In this section, we introduce notions of generalized partial fractional integrals and derivatives, in a multidimensional finite domain. They are natural generalizations of the corresponding fractional operators in the single variable case. Furthermore, similarly as in the integer order case, computation of partial fractional derivatives and integrals is reduced to the computation of one-variable fractional operators. Along the work, for i = 1, . . . , n, let a i , b i and α i be numbers in R and t = (t 1 , . . . , t n ) be such that t ∈ Ω n , where Ω n = (a 1 , b 1 ) × · · · × (a n , b n ) is a subset of R n . Moreover, let us define the following sets: . . , n. Let us assume that λ = (λ 1 , . . . , λ n ) and µ = (µ 1 , . . . , µ n ) are in R n . We shall present definitions of generalized partial fractional integrals and derivatives. Let Definition 2.5. The generalized partial fractional derivative of Riemann-Liouville type with respect to the ith variable t i is given by Definition 2.6. The generalized partial fractional derivative of Caputo type with respect to the ith variable t i is given by Example 3. Similarly, as in the one-dimensional case, partial operators K, A and B reduce to the standard partial fractional integrals and derivatives. The leftor right-sided Riemann-Liouville partial fractional integral with respect to the ith variable t i is obtained by choosing the kernel k α The standard left-and right-sided Riemann-Liouville and Caputo partial fractional derivatives with respect to ith variable t i are received by choosing the kernel k α . Moreover, one can easily check, that also variable order partial fractional integrals and derivatievs are particular cases of operators K Pi , A Pi and B Pi . Definitions of variable order partial fractional operators can be found in [18].
In classical calculus, integration by parts formula relates the integral of a product of functions to the integral of their derivative and antiderivative. As we can see below, this formula works also for generalized fractional operators, however it changes the type of differentiation: generalized partial fractional integrals K Pi are transformed into K P * i and generalized partial fractional Caputo derivatives B Pi are transformed into A P * i . In our setting, integration by parts changes a given parameter set P into its dual P * . The term duality comes from the fact that P * * = P . Definition 2.7 (Dual parameter set). Given a parameter set P = a, t, b, λ, µ we denote by P * the parameter set P * = a, t, b, µ, λ . We say that P * is the dual of P .
In the following, we shall write by dt = dt 1 . . . dt n and take arbitrary but fixed i ∈ {1, . . . , n}.
Theorem 2.8 (cf. [19]). Let P i = a i , t i , b i , λ i , µ i be the parameter set and let K Pi be the generalized partial fractional integral with k i being a difference kernel such that k i ∈ L 1 (0, b i − a i ; R). If f : R n → R and η : R n → R, f, η ∈ C Ω n ; R , then the generalized partial fractional integrals satisfy the following identity: where P * i is the dual of P i . Theorem 2.9 (cf. [19]). Let P i = a i , t i , b i , λ i , µ i be the parameter set and f, η ∈ C 1 Ω n ; R . Moreover, let B Pi = d dt • K Pi , where K Pi is the generalized partial fractional integral with difference kernel i.e., where ν i is the outward pointing unit normal to ∂Ω n .
3. Main results. In this section we study multidimensional generalized fractional variational problems. First we discuss the fundamental problem, for which we recall the necessary optimality condition and prove a sufficient condition for minimizer. Next, we consider the generalized fractional isoperimetric problem i.e., we want to minimize the generalized fractional variational functional subject to the boundary conditions and the generalized fractional isoperimetric constraint. We show the necessary optimality condition of the Euler-Lagrange type and, in particular cases, obtain results concerning constant and variable order fractional variational problems.
3.1. Fundamental problem. In order to state the generalized fundamental problem let us introduce the notion of a generalized fractional gradient.
Definition 3.1. Let n ∈ N and P = (P 1 , . . . , P n ), P i = a i , t , b i , λ i , µ i . We define a generalized fractional gradient of a function f : R n → R with respect to the generalized fractional operator T by where {e i : i = 1, . . . , n} denotes the standard basis in R n .
For n ∈ N let us assume that P i = a i , t i , b i , λ i , µ i and P = (P 1 , . . . , P n ), y : R n → R, and ζ : ∂Ω n → R is a given function. Consider the following functional: where A(ζ) := y ∈ C 1 (Ω n ; R) : y| ∂Ωn = ζ, K Pi [y], B Pi [y] ∈ C(Ω n ; R), i = 1, . . . , n , ∇ denotes the classical gradient operator, ∇ K P and ∇ B P are generalized fractional gradient operators such that K Pi is the generalized partial fractional integral with the kernel k i = k i (t i − τ ), k i ∈ L 1 (0, b i − a i ; R) and B Pi is the generalized partial fractional derivative of Caputo type satisfying B Pi = K Pi • ∂ ∂ti , for i = 1, . . . , n. Moreover, we assume that F is a Lagrangian of class C 1 : The next theorem states that if a function minimizes functional (2), then it necessarily must satisfy (3). This means that equation (3) determines candidates to solve problem of minimizing functional (2). Theorem 3.2 (cf. [19]). Suppose thatȳ ∈ A(ζ) is a minimizer of (2). Then,ȳ satisfies the following generalized Euler-Lagrange equation: The next theorem gives a sufficient condition for a minimizer of functional (2).
is convex for every t ∈Ω n . Then,ȳ is a minimizer of functional (2).

3.2.
Isoperimetric problem. Suppose that y : R n → R, P = a i , t i , b i , λ i , µ i , P = (P 1 , . . . , P n ) and ζ : ∂Ω n → R is a given curve. Let us define the following functional where operators ∇ K P ,∇, ∇ B P and function G are the same as in the case of functional (2).
be the Lagrange multiplier we find or, in other words, Finally, applying the integration by parts formula, and by the fundamental lemma of the calculus of variations we obtain (5).

Remark 1.
Notice that, if the main boundary condition y| ∂Ωn = ζ is not prescribed, then an extra boundary condition holds.

Moreover
• F, G are of class C 1 , Then, ifȳ is not an extremal for functional (8), we can find λ 0 ∈ R such that the following equation: Ifȳ ∈ C 1 (Ω n ; R) minimizes subject to (9) and [y] and derivative C ai D αi(·) ti are continuous onΩ n , and • F, G are of class C 1 , [∂ 1+2n+i G( y )(τ )] are continuously differentiable onΩ n , for i = 1, . . . , n. Then, there is λ 0 ∈ R such that, for H = F − λ 0 G,ȳ is a solution to the following equation: provided thatȳ is not a solution to Euler-Lagrange equation associated to J .
Example 4. Consider the problem of minimizing the following functional subject to the boundary condition y| ∂Ωn = ζ and an isoperimetric constraint J (y) = 0, where J : where ρ = ρ(t) > 0 is a weight function. Since ρ > 0 there is no solution to the Euler-Lagrange equation for functional J . The augmented Lagrangian is . , x n i ), i = 2, 3, 4, and by Theorem 3.4, the Euler-Lagrange equation for problem (10)-(11) is The next theorem gives sufficient condition for minimizer of functional (2) on the set A ξ (ζ).
Note that one can readily extend Theorem 3.4 to cope with abnormal problems: when the optimal solution is an extremal for the isoperimetric constraint. Because generalized isoperimetric problem can be related to a finite-dimensional optimization problem we can simply apply the abnormal version of the Lagrange multipliers introducing an additional multiplier λ 1 . Consequently the following result is true. Then, one can find real constants λ 0 and λ 1 , (λ 0 , λ 1 ) = 0, such that, for H = λ 1 F − λ 0 G, equation Ifȳ is not an extremal for J then we may take λ 1 = 1. Ifȳ is an extremal for J then we take λ 1 = 0, unlessȳ is also an extremal for I. In the latter case neither λ 0 nor λ 1 is determined.