Research article Special Issues

Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels

  • In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler-Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results.

    Citation: Fátima Cruz, Ricardo Almeida, Natália Martins. Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels[J]. AIMS Mathematics, 2021, 6(5): 5351-5369. doi: 10.3934/math.2021315

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  • In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler-Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results.



    The fractional calculus is an old subject and presents an extension of ordinary calculus [12,14]. It began at the same time with the works of Leibniz on differential calculus, where he questioned what could be a derivative of arbitrary real order α>0. Since then, a large number of definitions of fractional order integral and derivative operators have appeared. Thus, we find in the literature numerous works dealing with similar topics, but for different fractional operators. One possible way to avoid such issue is to consider a more general class of fractional operators, like, for example, fractional integrals and derivatives with arbitrary kernels [3,12] or other types of general fractional derivatives [16,17,18,19,20].

    Another possible approach to fractional calculus is, instead of fixing the fractional order α, the introduction of a new function that acts like a distribution of the orders of differentiation [9,10]. Our goal in this paper is to combine both ideas into a single operator, in order to obtain new results that will generalize some of the already known.

    The main purpose of this paper is to prove optimality conditions for variational problems that depend on distributed-order fractional derivatives with arbitrary kernels. Namely, we will prove the Euler-Lagrange equation and the natural boundary conditions for variational problems with and without integral constraints and also with an holonomic constraint. Moreover, we provide sufficient optimality conditions for all the variational problems studied in this paper. With this work we generalize several existent works on fractional calculus of variations such as [1,2,7,8,13].

    The structure of the paper is as follows. In Section 2, we recall the necessary definitions and results from fractional calculus that are needed to the present work. Our main contributions are presented in Section 3. We finalize the paper with some illustrative examples and concluding remarks.

    Throughout the paper, Γ represents the well-known Gamma function and [α] denotes the integer part of αR. We begin by recalling the definition of ψ-Riemann-Liouville fractional integrals of a function x of order αR+.

    Definition 2.1. [14] Let αR+, x:[a,b]R be an integrable function, and ψC1([a,b],R) another function with ψ(t)>0, for all t[a,b]. The left and right Riemann-Liouville fractional integrals of x with respect to the kernel ψ, of order α, are defined by

    Iα,ψa+x(t):=1Γ(α)taψ(τ)(ψ(t)ψ(τ))α1x(τ)dτ,for t>a,

    and

    Iα,ψbx(t):=1Γ(α)btψ(τ)(ψ(τ)ψ(t))α1x(τ)dτ,for t<b,

    respectively.

    Definition 2.2. [14] Let αR+, x:[a,b]R an integrable function, and ψCn([a,b],R) with ψ(t)>0, for all t[a,b]. The left and right Riemann-Liouville fractional derivatives of x with respect to the kernel ψ, of order α, are defined by

    Dα,ψa+x(t):=(1ψ(t)ddt)nInα,ψa+x(t)andDα,ψbx(t):=(1ψ(t)ddt)nInα,ψbx(t),

    respectively, where n=[α]+1.

    The operators Dα,ψa+x and Dα,ψbx can be simply called ψ-Riemann-Liouville fractional derivatives of x of order α [5]. If we interchange the order of the ordinary derivative with the fractional integral, we obtain the definition of the ψ-Caputo fractional derivatives of x of order α.

    Definition 2.3. [3] Given αR+, let nN be given by n=[α]+1 if αN, and n=α if αN. Given two functions x,ψCn([a,b],R) with ψ(t)>0, for all t[a,b], we define the left and right Caputo fractional derivatives of x with respect to the kernel ψ, of order α, by

    CDα,ψa+x(t):=Inα,ψa+(1ψ(t)ddt)nx(t)andCDα,ψbx(t):=Inα,ψb(1ψ(t)ddt)nx(t),

    respectively.

    Lemma 2.4. [3] Given α>0, let nN be given by Definition 2.3. For βR with β>n, we have that

    CDα,ψa+(ψ(t)ψ(a))β1=Γ(β)Γ(βα)(ψ(t)ψ(a))βα1

    and

    CDα,ψb(ψ(b)ψ(t))β1=Γ(β)Γ(βα)(ψ(b)ψ(t))βα1.

    Until the end of the work, the fractional order α belongs to the interval [0,1] and the kernel ψ is a function on the set C1([a,b],R), with ψ(t)>0, for all t[a,b].

    In order to introduce the new concepts of distributed-order fractional derivatives with respect to another function, in the Riemann-Liouville and in the Caputo sense, we consider a new continuous function ϕ:[0,1][0,1] that satisfies the condition

    10ϕ(α)dα>0.

    Usually, function ϕ is called order-weighting or strength function. For some applications on distributed-order fractional derivatives, we suggest the paper [11].

    Definition 2.5. Let x:[a,b]R be an integrable function. The left and right Riemann-Liouville distributed-order fractional derivatives of a function x with respect to the kernel ψ are defined by

    Dϕ(α),ψa+x(t):=10ϕ(α)Dα,ψa+x(t)dαandDϕ(α),ψbx(t):=10ϕ(α)Dα,ψbx(t)dα,

    where Dα,ψa+ and Dα,ψb are the left and right ψ-Riemann-Liouville fractional derivatives of order α, respectively.

    Definition 2.6. The left and right Caputo distributed-order fractional derivatives of a function xC1([a,b],R) with respect to the kernel ψ are defined by

    CDϕ(α),ψa+x(t):=10ϕ(α)CDα,ψa+x(t)dαandCDϕ(α),ψbx(t):=10ϕ(α)CDα,ψbx(t)dα,

    where CDα,ψa+ and CDα,ψb are the left and right ψ-Caputo fractional derivatives of order α, respectively.

    For our work we will also need the concepts of distributed-order fractional integrals with respect to the kernel ψ:

    I1ϕ(α),ψa+x(t):=10ϕ(α)I1α,ψa+x(t)dαandI1ϕ(α),ψbx(t):=10ϕ(α)I1α,ψbx(t)dα,

    where I1α,ψa+ and I1α,ψb are the left and right ψ-Riemann-Liouville fractional integrals of order 1α, respectively.

    In the sequel, let us consider two continuous functions ϕ,φ:[0,1][0,1] satisfying the conditions

    10ϕ(α)dα>0and10φ(α)dα>0.

    The goal of this work is to exhibit necessary and sufficient optimality conditions for the following fractional variational problem:

    Problem (P): Find a curve xC1([a,b],R) that minimizes or maximizes the following functional

    J(x):=baL(t,x(t),CDϕ(α),ψa+x(t),CDφ(α),ψbx(t))dt, (3.1)

    where L:[a,b]×R3R is assumed to be continuously differentiable with respect to the second, third and fourth variables. In our study, we will consider the variational problem with and without fixed boundary conditions, and also with an isoperimetric or holonomic constraints.

    Before proving our main results, we need to prove the following integration by parts formulae for the left and right Caputo distributed-order fractional derivatives with respect to another function.

    Theorem 3.1. (Integration by parts formulae) Let x:[a,b]R be a continuous function and y:[a,b]R a continuously differentiable function. Then,

    bax(t)CDϕ(α),ψa+y(t)dt=bay(t)(Dϕ(α),ψbx(t)ψ(t))ψ(t)dt+[y(t)(I1ϕ(α),ψbx(t)ψ(t))]t=bt=a

    and

    bax(t)CDφ(α),ψby(t)dt=bay(t)(Dφ(α),ψa+x(t)ψ(t))ψ(t)dt[y(t)(I1φ(α),ψa+x(t)ψ(t))]t=bt=a.

    Proof. By definition of the left ψ-Caputo distributed-order fractional derivative, we have the following:

    bax(t)CDϕ(α),ψa+y(t)dt=bax(t)10ϕ(α)CDα,ψa+y(t)dαdt=bax(t)10ϕ(α)Γ(1α)ta(ψ(t)ψ(τ))αy(τ)dτdαdt=10ϕ(α)Γ(1α)batax(t)(ψ(t)ψ(τ))αy(τ)dτdtdα.

    Reversing the order of integration, we get

    10ϕ(α)Γ(1α)batax(t)(ψ(t)ψ(τ))αy(τ)dτdtdα=10ϕ(α)Γ(1α)bay(τ)bτx(t)(ψ(t)ψ(τ))αdtdτdα.

    Using the standard integration by parts formula, we have

    10ϕ(α)Γ(1α)[bay(τ)bτx(t)(ψ(t)ψ(τ))αdtdτ]dα=10ϕ(α)Γ(1α)[[y(τ)bτx(t)(ψ(t)ψ(τ))αdt]τ=bτ=abay(τ)ddτ(bτx(t)(ψ(t)ψ(τ))αdt)dτ]dα.

    Therefore, one gets

    bax(t)CDϕ(α),ψa+y(t)dt=[y(τ)10ϕ(α)Γ(1α)bτψ(t)(ψ(t)ψ(τ))α(x(t)ψ(t))dtdα]τ=bτ=a+bay(τ)10ϕ(α)Γ(1α)(1ψ(τ)ddτ)(bτψ(t)(ψ(t)ψ(τ))α(x(t)ψ(t))dt)ψ(τ)dαdτ=[y(τ)10ϕ(α)(I1α,ψbx(τ)ψ(τ))dα]τ=bτ=a+bay(τ)10ϕ(α)(Dα,ψbx(τ)ψ(τ))dαψ(τ)dτ.

    Hence, we conclude that

    bax(t)CDϕ(α),ψa+y(t)dt=bay(t)(Dϕ(α),ψbx(t)ψ(t))ψ(t)dt+[y(t)(I1ϕ(α),ψbx(t)ψ(t))]t=bt=a

    as desired. Using similar techniques, we deduce the integration by parts formula involving the operator CDφ(α),ψb.

    In what follows, we will denote by iL the partial derivative of L with respect to its ith-coordinate and use the notation:

    [x](t):=(t,x(t),CDϕ(α),ψa+x(t),CDφ(α),ψbx(t)).

    We are now in a position to prove our first main result.

    Theorem 3.2. (Fractional Euler-Lagrange equation and natural boundary conditions) Let xC1([a,b],R) be a curve such that functional J as defined by (3.1) attains an extremum. If the maps

    tDϕ(α),ψb3L[x](t)ψ(t)andtDφ(α),ψa+4L[x](t)ψ(t)

    are continuous on [a,b], then x satisfies the following Euler-Lagrange equation

    2L[x](t)+(Dϕ(α),ψb3L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[x](t)ψ(t))ψ(t)=0, (3.2)

    for all t[a,b]. Also, if x(a) is free, then

    I1ϕ(α),ψb3L[x](t)ψ(t)=I1φ(α),ψa+4L[x](t)ψ(t),att=a, (3.3)

    and if x(b) is free, then

    I1ϕ(α),ψb3L[x](t)ψ(t)=I1φ(α),ψa+4L[x](t)ψ(t),att=b. (3.4)

    Proof. Let hC1([a,b],R) be an arbitrary function and define the function j by j(ϵ):=J(x+ϵh), ϵR. Since x is an extremizer of J, j(0)=0, and, therefore,

    ba(2L[x](t)h(t)+3L[x](t)CDϕ(α),ψa+h(t)+4L[x](t)CDφ(α),ψbh(t))dt=0.

    Using Theorem 3.1 we obtain

    ba(2L[x](t)+(Dϕ(α),ψb3L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[x](t)ψ(t))ψ(t))h(t)dt+[h(t)(I1ϕ(α),ψb3L[x](t)ψ(t))]t=bt=a[h(t)(I1φ(α),ψa+4L[x](t)ψ(t))]t=bt=a=0. (3.5)

    If we restrict the variations h by considering h(a)=h(b)=0, we have

    ba(2L[x](t)+(Dϕ(α),ψb3L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[x](t)ψ(t))ψ(t))h(t)dt=0.

    Since h is arbitrary, from the Fundamental Lemma of Calculus of Variations (see [15]), we get

    2L[x](t)+(Dϕ(α),ψb3L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[x](t)ψ(t))ψ(t)=0,

    for all t[a,b], proving the Euler-Lagrange equation (3.2). If x(a) is free, considering h(a)0 and h(b)=0 in (3.5) and using (3.2), we obtain

    h(a)(I1φ(α),ψa+4L[x]ψ(a)I1ϕ(α),ψb3L[x]ψ(a))=0.

    Since h(a) is arbitrary, we get that, at t=a,

    I1ϕ(α),ψb3L[x](t)ψ(t)=I1φ(α),ψa+4L[x](t)ψ(t),

    proving the natural boundary condition (3.3). Similarly, if x(b) is free, considering h(a)=0 and h(b)0 in (3.5) and using (3.2), we deduce the natural boundary condition (3.4).

    Remark 1. It is clear that the variational problem (P) can be easily extended to functionals depending on a vector function x:=(x1,...,xn). More precisely, let L:[a,b]×R3nR be a continuously differentiable function with respect to the j-th variable, for j=2,,3n+1, and consider the functional

    J(x):=baL[x](t)dt.

    It follows from the proof of Theorem 3.2 that, if functional J attains an extremum at x=(x1,...,xn), then, for all t[a,b] and i=1,...,n,

    i+1L[x](t)+(Dϕ(α),ψbi+n+1L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+i+2n+1L[x](t)ψ(t))ψ(t)=0.

    If the state values x(a) and x(b) are free, then we get the following 2n natural boundary conditions:

    I1ϕ(α),ψbi+n+1L[x](t)ψ(t)=I1φ(α),ψa+i+2n+1L[x](t)ψ(t)at t=a

    and

    I1ϕ(α),ψbi+n+1L[x](t)ψ(t)=I1φ(α),ψa+i+2n+1L[x](t)ψ(t)at t=b,

    for all i=1,...,n.

    Next, we consider problem (P) subject to an integral constraint of type

    I(x):=baG[x](t)dt=k, (3.6)

    where kR is fixed and G:[a,b]×R3R is a continuously differentiable function with respect to the second, third and fourth variables. This type of problems are known in the literature as isoperimetric problems.

    Theorem 3.3. (Necessary optimality conditions for isoperimetric problems I) Let x be a curve such that J attains an extremum at x, when subject to the integral constraint (3.6). Assume that x does not satisfies the equation

    2G[x](t)+(Dϕ(α),ψb3G[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4G[x](t)ψ(t))ψ(t)=0,t[a,b]. (3.7)

    If the maps

    t(Dϕ(α),ψb3L[x](t)ψ(t)),t(Dφ(α),ψa+4L[x](t)ψ(t)),t(Dϕ(α),ψb3G[x](t)ψ(t))andt(Dφ(α),ψa+4G[x](t)ψ(t))

    are continuous on [a,b], then there exists a real number λ such that x is a solution of the equation

    2H[x](t)+(Dϕ(α),ψb3H[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4H[x](t)ψ(t))ψ(t)=0, (3.8)

    for all t[a,b], where H:=L+λG. Also, if the state variable x(a) is free, then

    I1ϕ(α),ψb3H[x](t)ψ(t)=I1φ(α),ψa+4H[x](t)ψ(t)att=a, (3.9)

    and if x(b) is free, then x must satisfy

    I1ϕ(α),ψb3H[x](t)ψ(t)=I1φ(α),ψa+4H[x](t)ψ(t)att=b. (3.10)

    Proof. Suppose that x is an extremizer of functional J subject to the integral constraint (3.6). Let h1,h2C1([a,b],R) be two functions. First, suppose that hi(a)=hi(b)=0, for i=1,2, and define the two functions i and j in the following way

    i(ϵ1,ϵ2):=I(x+ϵ1h1+ϵ2h2)kandj(ϵ1,ϵ2):=J(x+ϵ1h1+ϵ2h2),

    for ϵ1,ϵ2R. Using similar techniques as the ones used in the proof of Theorem 3.2, we get

    2i(0,0)=ba(2G[x](t)+(Dϕ(α),ψb3G[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4G[x](t)ψ(t))ψ(t))h2(t)dt+[(I1ϕ(α),ψb3G[x](t)ψ(t))h2(t)]t=bt=a[(I1φ(α),ψa+4G[x](t)ψ(t))h2(t)]t=bt=a.

    Since h2(a)=h2(b)=0, we conclude that

    2i(0,0)=ba(2G[x](t)+(Dϕ(α),ψb3G[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4G[x](t)ψ(t))ψ(t))h2(t)dt.

    Since x does not satisfies equation (3.7), one concludes that there exists t0[a,b] such that,

    2G[x](t0)+(Dϕ(α),ψb3G[x](t0)ψ(t0))ψ(t0)+(Dφ(α),ψa+4G[x](t0)ψ(t0))ψ(t0)0.

    Then, there exists some function h2 for which 2i(0,0)0. Also, i(0,0)=0 and so, applying the Implicit Function Theorem, we conclude that there exists a continuously differentiable function g defined on an open set UR containing 0, such that g(0)=0 and i(ϵ1,g(ϵ1))=0, for all ϵ1U. Hence, there exists an infinity subfamily of functions x+ϵ1h1+g(ϵ1)h2 that satisfies the integral restriction (3.6). From now on we will consider such subfamily of variations. Observe that the vector (0,0) is an extremizer of j, subject to the constraint i(,)=0. Since i(0,0)(0,0), by the Lagrange Multiplier Rule, there exists a real number λ such that (j+λi)(0,0)=(0,0). Hence, 1(j+λi)(0,0)=0, and, therefore,

    ba(2L[x](t)+(Dϕ(α),ψb3L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[x](t)ψ(t))ψ(t)+λ(2G[x](t)+(Dϕ(α),ψb3G[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4G[x](t)ψ(t))ψ(t)))h1(t)dt+[(I1ϕ(α),ψb3L[x](t)ψ(t))h1(t)]t=bt=a[(I1φ(α),ψa+4L[x](t)ψ(t))h1(t)]t=bt=a+λ[(I1ϕ(α),ψb3G[x](t)ψ(t))h1(t)]t=bt=aλ[(I1φ(α),ψa+4G[x](t)ψ(t))h1(t)]t=bt=a=0. (3.11)

    Since h1 is an arbitrary function and considering h1(a)=h1(b)=0, it follows from the Fundamental Lemma of Calculus of Variations that

    2L[x](t)+(Dϕ(α),ψb3L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[x](t)ψ(t))ψ(t)+λ(2G[x](t)+(Dϕ(α),ψb3G[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4G[x](t)ψ(t))ψ(t))=0,

    for all t[a,b], proving equation (3.8).

    Suppose now that x(a) is free and consider variations h1 with h1(a)0 and h1(b)=0. From (3.11) and using (3.8), we conclude that

    h1(a)(I1ϕ(α),ψb3H[x]ψ(a)I1φ(α),ψa+4H[x]ψ(a))=0,

    proving (3.9). Similarly, if x(b) is free, then by considering h1(a)=0 and h1(b)0 in (3.11) and using (3.8), (3.10) is proved.

    Theorem 3.4. (Necessary optimality conditions for isoperimetric problems II) Let x be a curve such that J attains an extremum at x, when subject to the integral constraint (3.6). If the maps

    t(Dϕ(α),ψb3L[x](t)ψ(t)),t(Dφ(α),ψa+4L[x](t)ψ(t)),t(Dϕ(α),ψb3G[x](t)ψ(t))andt(Dφ(α),ψa+4G[x](t)ψ(t))

    are continuous on [a,b], then there exists a vector (λ0,λ)R2{(0,0)} such that x is a solution of the equation (3.8) for all t[a,b], with the Hamiltonian H defined as H:=λ0L+λG. Also, if the state variable x(a) is free, then x must satisfy the equation (3.9) and if x(b) is free, then x must satisfy the equation (3.10).

    Proof. The proof is similar to the one of Theorem 3.3. Since the vector (0,0) is an extremizer of j, subject to the constraint i(,)=0, the Lagrange Multiplier Rule guarantees the existence of two constants λ0,λR, not both zero, such that (λ0j+λi)(0,0)=(0,0). Computing 1(λ0j+λi)(0,0)=0, we obtain the desired result.

    Now, we consider problem (P) but in presence of an holonomic restriction. Suppose that the state variable x is a two-dimensional vector function x=(x1,x2), where x1,x2C1([a,b],R). We impose the following restriction to our variational problem:

    g(t,x(t))=0,t[a,b], (3.12)

    where g:[a,b]×R2R is a continuously differentiable function. Also, boundary conditions

    x(a)=xaandx(b)=xb,xa,xbR2 (3.13)

    may be imposed to the variational problem.

    Theorem 3.5. (Necessary optimality conditions for variational problems with an holonomic constraint) Consider the functional

    J(x)=baL[x](t)dt, (3.14)

    defined on C1([a,b],R)×C1([a,b],R) and subject to the constraint (3.12). If x is an extremizer of functional J, if the maps

    t(Dϕ(α),ψbi+3L[x](t)ψ(t))andt(Dφ(α),ψa+i+5L[x](t)ψ(t)),i=1,2,

    are continuous, and if

    3g(t,x(t))0,t[a,b],

    then there exists a continuous function λ:[a,b]R such that x is a solution of

    i+1L[x](t)+(Dϕ(α),ψbi+3L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+i+5L[x](t)ψ(t))ψ(t)+λ(t)i+1g(t,x(t))=0,t[a,b],i=1,2. (3.15)

    Also, if x(a) is free, then, for i=1,2,

    I1ϕ(α),ψbi+3L[x](t)ψ(t)=I1φ(α),ψa+i+5L[x](t)ψ(t)att=a (3.16)

    and if x(b) is free, then, for i=1,2,

    I1ϕ(α),ψbi+3L[x](t)ψ(t)=I1φ(α),ψa+i+5L[x](t)ψ(t)att=b. (3.17)

    Proof. Let h=(h1,h2)C1([a,b],R)×C1([a,b],R). To prove Eqs (3.15), first assume that h(a)=(0,0)=h(b) and let ϵR. Since the variations must fulfill the holonomic restriction (3.12), then

    g(t,x1(t)+ϵh1(t),x2(t)+ϵh2(t))=0,t[a,b]. (3.18)

    Differentiating (3.18) with respect to ϵ and taking ϵ=0, we conclude that

    3g(t,x(t))h2(t)=2g(t,x(t))h1(t),t[a,b]. (3.19)

    Define the function λ:[a,b]R by the rule

    λ(t):=3L[x](t)+(Dϕ(α),ψb5L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+7L[x](t)ψ(t))ψ(t)3g(t,x(t)). (3.20)

    From the definition of λ, we prove equation (3.15) for i=2. Now, we prove that equation (3.15) holds for i=1. By Eqs (3.19) and (3.20) we obtain

    λ(t)2g(t,x(t))h1(t)=(3L[x](t)+(Dϕ(α),ψb5L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+7L[x](t)ψ(t))ψ(t))h2(t). (3.21)

    Let us define the new function j by the rule j(ϵ):=J(x1+ϵh1,x2+ϵh2), ϵR. Since j(0)=0, we conclude that

    ba(2L[x](t)+(Dϕ(α),ψb4L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+6L[x](t)ψ(t))ψ(t))h1(t)dt+[(I1ϕ(α),ψb4L[x](t)ψ(t))h1(t)]t=bt=a[(I1φ(α),ψa+6L[x](t)ψ(t))h1(t)]t=bt=a+ba(3L[x](t)+(Dϕ(α),ψb5L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+7L[x](t)ψ(t))ψ(t))h2(t)dt+[(I1ϕ(α),ψb5L[x](t)ψ(t))h2(t)]t=bt=a[(I1φ(α),ψa+7L[x](t)ψ(t))h2(t)]t=bt=a=0 (3.22)

    and by considering h(a)=h(b)=(0,0), we obtain

    ba((2L[x](t)+(Dϕ(α),ψb4L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+6L[x](t)ψ(t))ψ(t))h1(t)+(3L[x](t)+(Dϕ(α),ψb5L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+7L[x](t)ψ(t))ψ(t))h2(t))dt=0.

    Using Eq (3.21), we obtain

    ba(2L[x](t)+(Dϕ(α),ψb4L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+6L[x](t)ψ(t))ψ(t)+λ(t)2g(t,x(t)))h1(t)dt=0.

    Since h1 is arbitrary, from the Fundamental Lemma of Calculus of Variations, we get

    2L[x](t)+(Dϕ(α),ψb4L[x](t)ψ(t))ψ(t)+(Dφ(α),ψa+6L[x](t)ψ(t))ψ(t)+λ(t)2g(t,x(t))=0,

    for all t[a,b], proving Eq (3.15) for i=1. We now prove the transversality conditions (3.16) and (3.17). If x(a) is free, then by considering h(a)(0,0) and h(b)=(0,0) in (3.22) and using (3.15), (3.16) is proved. If x(b) is free, then consider h(a)=(0,0) and h(b)(0,0) to deduce (3.17).

    Now we will prove sufficient optimality conditions for all the variational problems studied in the last subsection.

    Definition 3.6. We say that f(t,x2,x3,...,xn) is a convex (resp. concave) function in URn if if(t,x2,x3,...,xn), i=2,,n, exist and are continuous, and if

    f(t,x2+η2,x3+η3,...,xn+ηn)f(t,x2,x3,...,xn)(resp. )ni=2if(t,x2,x3,...,xn)ηi

    for all (t,x2,x3,...,xn),(t,x2+η2,x3+η3,...,xn+ηn)U.

    Theorem 3.7. (Sufficient optimality conditions) Suppose that the Lagrangian function L is convex (resp. concave) in [a,b]×R3. Then, each solution ¯x of the fractional Euler-Lagrange equation (3.2) minimizes (resp. maximizes) the functional J given in (3.1), subject to the boundary conditions x(a)=¯x(a) and x(b)=¯x(b). Also, if x(a) is free, then each solution ¯x of the equations (3.2) and (3.3) minimizes (resp. maximizes) J. If x(b) is free, then each solution ¯x of the equations (3.2) and (3.4) minimizes (resp. maximizes) J.

    Proof. Let ηC1([a,b],R) be an arbitrary function. Since L is convex, we have

    J(¯x+η)J(¯x)ba(2L[¯x](t)η(t)+3L[¯x](t)CDϕ(α),ψa+η(t)+4L[¯x](t)CDφ(α),ψbη(t))dt.

    Applying Theorem 3.1, we obtain

    J(¯x+η)J(¯x)ba(2L[¯x](t)+(Dϕ(α),ψb3L[¯x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[¯x](t)ψ(t))ψ(t))η(t)dt+[(I1ϕ(α),ψb3L[¯x](t)ψ(t))η(t)]t=bt=a[(I1φ(α),ψa+4L[¯x](t)ψ(t))η(t)]t=bt=a. (3.23)

    If x(a) and x(b) are fixed, then the admissible variations must fulfill the conditions η(a)=η(b)=0, and so we get

    J(¯x+η)J(¯x)ba(2L[¯x](t)+(Dϕ(α),ψb3L[¯x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[¯x](t)ψ(t))ψ(t))η(t)dt=0,

    since ¯x is a solution of the fractional Euler-Lagrange equation (3.2). If x(a) is free, then by considering η(a)0 and η(b)=0 in (3.23), we have

    J(¯x+η)J(¯x)ba(2L[¯x](t)+(Dϕ(α),ψb3L[¯x](t)ψ(t))ψ(t)+(Dφ(α),ψa+4L[¯x](t)ψ(t))ψ(t))η(t)dt+η(a)(I1ϕ(α),ψb3L[¯x]ψ(a)+I1φ(α),ψa+4L[¯x]ψ(a))=0,

    since ¯x is a solution of the fractional equations (3.2) and (3.3). Similarly, if x(b) is free, then by considering η(a)=0 and η(b)0 in (3.23), since ¯x is a solution of the fractional equations (3.2) and (3.4), we conclude that J(¯x+η)J(¯x)0. The cases when L is concave are proven in a similar way.

    Theorem 3.8. (Sufficient optimality conditions for isoperimetric problems) Let us assume that, for some constant λ, the functions L and λG are convex (resp. concave) in [a,b]×R3 and define the function H as H=L+λG. Then, each solution ¯x of the fractional equation (3.8) minimizes (resp. maximizes) the functional J given in (3.1), subject to the restrictions x(a)=¯x(a) and x(b)=¯x(b), and the integral constraint (3.6). Also, if x(a) is free, then each solution ¯x of the fractional equations (3.8) and (3.9) minimizes (resp. maximizes) J subject to (3.6). If x(b) is free, then each solution ¯x of the fractional equations (3.8) and (3.10) minimizes (resp. maximizes) J subject to (3.6).

    Proof. First, assume that functions L and λG are convex. It is easy to verify that function H is convex. Let ηC1([a,b],R) be such that η(a)=η(b)=0. By Theorem 3.7, ¯x minimizes ˜H:=ba(L+λG)dt, that is, ˜H(¯x+η)˜H(¯x). So, if xC1([a,b],R) is any function such that x(a)=¯x(a) and x(b)=¯x(b), then

    baL[x](t)dt+λbaG[x](t)dtbaL[¯x](t)+λbaG[¯x](t)dt.

    If we restrict to the integral constraint, we have

    baL[x](t)dt+λkbaL[¯x](t)dt+λk.

    Therefore,

    baL[x](t)dtbaL[¯x](t)dt,

    this is, J(x)J(¯x). The remaining cases are proven in a similar way.

    Theorem 3.9. (Sufficient optimality conditions for variational problems with an holonomic constraint) Consider the functional J defined in (3.14), where the Lagrangian function L is convex (resp. concave) in [a,b]×R6, and function λ:[a,b]R given by formula (3.20). Then, each solution ¯x=(¯x1,¯x2) of equations (3.15) minimizes (resp. maximizes) the functional J, subject to the constraints x(a)=¯x(a) and x(b)=¯x(b), and the holonomic restriction (3.12). Also, if x(a) is free, then each solution ¯x of the fractional equations (3.15) and (3.16) minimizes (resp. maximizes) J subject to (3.12). If x(b) is free, then each solution ¯x of the fractional equations (3.15) and (3.17) minimizes (resp. maximizes) J subject to (3.12).

    Proof. We shall give the proof only for the case where L is convex; the concave case is analogous. Let η1,η2C1([a,b],R) be arbitrary functions, where η=(η1,η2) is a differentiable function with η(a)=(0,0)=η(b). Since L is convex, we have

    J(¯x+η)J(¯x)ba(2L[¯x](t)η1(t)+3L[¯x](t)η2(t)+4L[¯x](t)CDϕ(α),ψa+η1(t)+5L[¯x](t)CDϕ(α),ψa+η2(t)+6L[¯x](t)CDφ(α),ψbη1(t)+7L[¯x](t)CDφ(α),ψbη2(t))dt.

    Using the integration by parts formulae, we obtain

    J(¯x+η)J(¯x)ba(2L[¯x](t)+(Dϕ(α),ψb4L[¯x](t)ψ(t))ψ(t)+(Dφ(α),ψa+6L[¯x](t)ψ(t))ψ(t))η1(t)dt+ba(3L[¯x](t)+(Dϕ(α),ψb5L[¯x](t)ψ(t))ψ(t)+(Dφ(α),ψa+7L[¯x](t)ψ(t))ψ(t))η2(t)dt+[(I1ϕ(α),ψb4L[¯x](t)ψ(t))η1(t)]t=bt=a[(I1φ(α),ψa+6L[¯x](t)ψ(t))η1(t)]t=bt=a+[(I1ϕ(α),ψb5L[¯x](t)ψ(t))η2(t)]t=bt=a[(I1φ(α),ψa+7L[¯x](t)ψ(t))η2(t)]t=bt=a.

    Using equations (3.19) and (3.20), we get

    J(¯x+η)J(¯x)ba(2L[¯x](t)+(Dϕ(α),ψb4L[¯x](t)ψ(t))ψ(t)+(Dφ(α),ψa+6L[¯x](t)ψ(t))ψ(t)+λ(t)2g(t,¯x(t)))η1(t)dt+[(I1ϕ(α),ψb4L[¯x](t)ψ(t))η1(t)]t=bt=a[(I1φ(α),ψa+6L[¯x](t)ψ(t))η1(t)]t=bt=a+[(I1ϕ(α),ψb5L[¯x](t)ψ(t))η2(t)]t=bt=a[(I1φ(α),ψa+7L[¯x](t)ψ(t))η2(t)]t=bt=a. (3.24)

    Since η(a)=(0,0)=η(b), we obtain

    J(¯x+η)J(¯x)ba(2L[¯x](t)+(Dϕ(α),ψb4L[¯x](t)ψ(t))ψ(t)+(Dφ(α),ψa+6L[¯x](t)ψ(t))ψ(t)+λ(t)2g(t,¯x(t)))η1(t)dt=0,

    since ¯x is a solution of the fractional Euler-Lagrange equation (3.15) for i=1, for all t[a,b]. If x(a) is free, then by considering η(a)(0,0) and η(b)=(0,0) in (3.24), we get

    J(¯x+η)J(¯x)ba(2L[¯x](t)+(Dϕ(α),ψb4L[¯x](t)ψ(t))ψ(t)+(Dφ(α),ψa+6L[¯x](t)ψ(t))ψ(t)+λ(t)2g(t,¯x(t)))η1(t)dt+(I1φ(α),ψa+6L[¯x]ψ(a)I1ϕ(α),ψb4L[¯x]ψ(a))η1(a)+(I1φ(α),ψa+7L[¯x]ψ(a)I1ϕ(α),ψb5L[¯x]ψ(a))η2(a)=0,

    since ¯x is a solution of the fractional equations (3.15) and (3.16). Similarly, if x(b) is free, then by considering η(a)=(0,0) and η(b)(0,0) in (3.24), and since ¯x is a solution of the fractional equations (3.15) and (3.17), we conclude that J(¯x+η)J(¯x)0, proving the desired result.

    In this section we provide three examples in order to illustrate our results.

    Example 1. Suppose we want to minimize the following functional

    J(x)=10((x(t)(ψ(t)ψ(0))4)2+(CDϕ(α),ψ0+x(t)+(ψ(t)ψ(0))3(ψ(t)ψ(0))4ln(ψ(t)ψ(0)))2)dt,

    in the class of functions C1([0,1],R) subject to the restriction x(0)=0, where ϕ:[0,1][0,1] is defined by ϕ(α)=Γ(5α)4!. From Theorem 3.2, every local extremizer x of functional J such that

    tDϕ(α),ψ13L[x](t)ψ(t) (4.1)

    is continuous on [0,1], satisfies the following necessary conditions

    x(t)(ψ(t)ψ(0))4+(Dϕ(α),ψ1CDϕ(α),ψ0+x(t)+(ψ(t)ψ(0))3(ψ(t)ψ(0))4ln(ψ(t)ψ(0))ψ(t))ψ(t)=0, (4.2)

    for all t[0,1] and, at t=1,

    I1ϕ(α),ψ1CDϕ(α),ψ0+x(t)+(ψ(t)ψ(0))3(ψ(t)ψ(0))4ln(ψ(t)ψ(0))ψ(t)=0. (4.3)

    Note that the function ¯x:[0,1]R defined by ¯x(t)=(ψ(t)ψ(0))4 is such that

    CDα,ψ0+¯x(t)=4!Γ(5α)(ψ(t)ψ(0))4α(by Lemma 2.4).

    Thus,

    CDϕ(α),ψ0+¯x(t)=(ψ(t)ψ(0))3+(ψ(t)ψ(0))4ln(ψ(t)ψ(0)),

    and therefore ¯x satisfies condition (4.1), the Euler-Lagrange equation (4.2), and the natural boundary condition (4.3). Since the Lagrangian function is convex, by Theorem 3.7, we conclude that ¯x is a minimizer of J.

    Example 2. Suppose we want to minimize the following functional

    J(x)=10(t2+((ψ(1)ψ(t))αCDφ(α),ψ1x(t)(ψ(1)ψ(t))α+1)2)dt,

    in the class of functions C1([0,1],R) subject to the restriction x(1)=0, where φ:[0,1][0,1] is defined by φ(α)=2Γ(α+2). From Theorem 3.2, every local extremizer x of functional J such that

    tDφ(α),ψ0+4L[x](t)ψ(t) (4.4)

    is continuous on [0,1], satisfies the following necessary conditions

    (Dφ(α),ψ0+((ψ(1)ψ(t))αCDφ(α),ψ1x(t)(ψ(1)ψ(t))α+1)(ψ(1)ψ(t))αψ(t))ψ(t)=0, (4.5)

    for all t[0,1] and

    I1φ(α),ψ0+((ψ(1)ψ(t))αCDφ(α),ψ1x(t)(ψ(1)ψ(t))α+1)(ψ(1)ψ(t))αψ(t)=0,att=0. (4.6)

    Note that, by Lemma 2.4, if ¯x:[0,1]R is defined by ¯x(t)=(ψ(1)ψ(t))α+12, then

    CDα,ψ1¯x(t)=Γ(α+2)2(ψ(1)ψ(t)).

    Thus,

    CDφ(α),ψ1¯x(t)=ψ(1)ψ(t),

    and therefore ¯x satisfies condition (4.4), the Euler-Lagrange equation (4.5), and the natural boundary condition (4.6). Since the Lagrangian function is convex, by Theorem 3.7, ¯x is indeed a minimizer of J.

    Example 3. Consider now the following problem

    J(x)=10(x2(t)+(ψ(1)ψ(t))6α+2+(CDφ(α),ψ1x(t))2+14((ψ(1)ψ(t))3ψ(1)+ψ(t)ln(ψ(1)ψ(t)))2)dtmin,

    in the class of functions C1([0,1],R), subject to the restriction x(1)=0 and to the integral constraint

    I(x)=10(x(t)(ψ(1)ψ(t))3α+1+12((ψ(1)ψ(t))3ψ(1)+ψ(t)ln(ψ(1)ψ(t)))CDφ(α),ψ1x(t))dt=k,

    where

    k=10(14((ψ(1)ψ(t))3ψ(1)+ψ(t)ln(ψ(1)ψ(t)))2+(ψ(1)ψ(t))6α+2)dt,

    and φ:[0,1][0,1] is defined by φ(α)=Γ(2α+2)Γ(3α+2). Consider the function ¯x:[0,1]R defined by ¯x(t)=(ψ(1)ψ(t))3α+1. Then, by Lemma 2.4,

    CDα,ψ1¯x(t)=Γ(3α+2)Γ(2α+2)(ψ(1)ψ(t))2α+1

    and so

    CDφ(α),ψ1¯x(t)=12((ψ(1)ψ(t))3ψ(1)+ψ(t)ln(ψ(1)ψ(t))).

    Let

    H:=(x(t)(ψ(1)ψ(t))3α+1)2+(CDφ(α),ψ1x(t)12((ψ(1)ψ(t))3ψ(1)+ψ(t)ln(ψ(1)ψ(t))))2.

    Therefore, ¯x satisfies the Euler-Lagrange equation with respect to the Hamiltonian H:

    (x(t)(ψ(1)ψ(t))3α+1)+(Dφ(α),ψ0+CDφ(α),ψ1x(t)12((ψ(1)ψ(t))3ψ(1)+ψ(t)ln(ψ(1)ψ(t)))ψ(t))ψ(t)=0, (4.7)

    for all t[0,1] and the transversality condition

    I1φ(α),ψ0+CDφ(α),ψ1x(t)12((ψ(1)ψ(t))3ψ(1)+ψ(t)ln(ψ(1)ψ(t)))ψ(t)=0,t=0. (4.8)

    Thus, ¯x satisfies the necessary conditions of Theorem 3.3 with λ=2. Since the Hamiltonian function H is convex, by Theorem 3.8, a solution of equations (4.7) and (4.8) is actually a minimizer of J subject to the previous integral constraint. Hence,

    ¯x(t)=(ψ(1)ψ(t))3α+1

    is a solution of the proposed problem.

    In this work we generalized some of the results presented in [4] and [6], by considering in the Lagrangian functional a new fractional derivative that combines the two ones given in those papers. Namely, we deduced necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions.

    For future, we intend to generalize the results presented in this paper, by considering variational problems with higher-order derivatives and delayed arguments. Also, we intend to study variational problems of Herglotz type involving the new distributed-order fractional derivatives with arbitrary kernels introduced in this paper.

    This work is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), reference UIDB/04106/2020.

    All authors declare no conflicts of interest in this paper.



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