Variational problems for Holderian functions with free terminal point
aa r X i v : . [ m a t h . D S ] M a y Variational problems for H¨olderian functions with freeterminal point
Ricardo Almeida [email protected]
Nat´alia Martins [email protected]
CIDMA – Center for Research and Development in Mathematics and Applications,Department of Mathematics, University of Aveiro, Portugal.
Abstract
We develop the new variational calculus introduced in 2011 by J. Cresson and I. Gr-eff, where the classical derivative is substituted by a new complex operator called the scalederivative. In this paper we consider several nondifferentiable variational problems with freeterminal point: with and without constraints, of first and higher-order type.
Keywords:
H¨olderian functions; calculus of variations; Euler-Lagrange equation; naturalboundary conditions.
Mathematics Subject Classification 2010: primary: 39A13; 49K05; 49S05; secondary:26A27; 26B20; 49K10
Dating back to the late 17th century, the calculus of variations has been proved to be a powerfulltool in several fields, such as physics, geometry, engineering, economics and control theory. One ofthe earliest variational problems posed in physics was the problem of determining the shape of asurface of revolution that would encounter the least resistance when moving through some resistingmedium. This problem was solved by Issac Newton and the solution was published in 1687 in thefirst book of the collection
Principia . Since then, important mathematicians and physicists, suchas John and James Bernoulli, Leibniz, Euler, Lagrange, Jacobi, Weierstrass, Hilbert, Noether,Tonelly and Lebesgue, studied different variational problems, contributing to the development ofthe classical calculus of variations.The fundamental problem of the classical calculus of variations can be formulated in the fol-lowing form: among all differentiable functions y : [ a, b ] → R such that y ( a ) = y a and y ( b ) = y b ,where y a , y b are fixed real numbers, find the ones that minimize (or maximize) the functional L [ y ] = Z ba L ( t, y ( t ) , y ′ ( t )) dt. It can be proved that the extremizers of this variational problem must satisfy the following second-order differential equation ddt ∂L∂v ( t, y ( t ) , y ′ ( t )) = ∂L∂y ( t, y ( t ) , y ′ ( t )) , ∀ t ∈ [ a, b ]called the Euler-Lagrange equation (where ∂L∂y and ∂L∂v denote, respectively, the partial derivativeof L with respect to the second and third argument). With the two boundary restrictions, y ( a ) = y a and y ( b ) = y b , we may determine the extremals of L . If the boundary condition y ( a ) = y a is not1mposed in the initial problem, then in order to find the extremizers we have to add anothernecessary condition: ∂L∂v ( a, y ( a ) , y ′ ( a )) = 0; and thus, instead of the initial condition y ( a ) = y a ,we have one more equation on the system. Also, if y ( b ) = y b is not fixed in the problem, thenwe need to impose another necessary condition: ∂L∂v ( b, y ( b ) , y ′ ( b )) = 0. These two conditions areusually called natural boundary conditions or tranversality conditions.Since many physical phenomena are described by nondifferentiable functions, and many prob-lems posed in physics can be formulated using integral functionals, it seemed clear for somemathematicians and physicists that it would be interesting to develop a calculus of variations forfunctionals which are defined on a set of nondifferentiable functions.There exist several different approaches to deal with nondifferentiability in problems of thecalculus of variations: the time scale approach, which typically deals with delta or nabla differen-tiable functions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]; the fractional approach, which deals with fractionalderivatives of order less than one [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]; the quantum approach,which deals with quantum derivatives [21, 22, 23, 24, 25, 26]; and the scale derivative approachrecently introduced by J. Cresson in 2005 [27].In this paper we are concerned with the scale derivative approach. In the paper [29], J. Cressonand I. Greff introduced a new variational calculus where the classical derivative is replaced by a newcomplex operator called the scale derivative. Basically, this new derivative allows the developmentof an analogue of a differential variational calculus for H¨olderian functions. The scale variationaltheory is still in the very beginning [27, 28, 29, 30, 31, 32], and much remains to be done.The main results of the paper [29] are: the Euler-Lagrange equation for a variational problemwith fixed boundary conditions involving a scale derivative, and the nondifferentiable Noether’sTheorem. In this work we will use the scale derivative introduced in [29] (we remark that suchderivative is not exactly the same as introduced in [27]).In [32] the following scale variational problems were studied: the isoperimetric problem, thevariational problem with dependence on a parameter, the higher-order variational problem, and thevariational problem with two independent variables, for problems with fixed boundary conditions.The main purpose of this paper is to generalize the results of [29, 32]. Let a and b be two fixedreal numbers such that a ≤ b . We consider nondifferentiable variational problems I [ y, T ] = Z Ta L (cid:18) t, y ( t ) , (cid:3) y (cid:3) t ( t ) (cid:19) dt where a ≤ T ≤ b . The admissible functions y are H¨olderian such that y ( a ) = y a , for some fixed y a ∈ R , and (cid:3) y/ (cid:3) t is the scale derivative of y (to be defined later). Note that here, in contrastto [29, 32], we have a free terminal point T and no constraint is imposed on y ( T ). Therefore, T and y ( T ) become part of the extremal choice process.In this paper, for expository convenience, we assume that only the terminal point is variable.It is clear that all the arguments used in the proofs of our results are easily extended to the caseof a variable initial point.This paper is organized as follows. In Section 2 we present the scale derivative as introduced in[29] and briefly review some of its properties, namely, the Leibniz rule and the Barrow rule in thescale calculus context. Our main results are presented in Section 3. In Subsection 3.1 we provenecessary and sufficient conditions to obtain extremals for complex valued integral functionals withvarious type of constraint (imposed over the terminal point T and/or over y ( T )). In Subsection3.2 we briefly show how higher-order scale derivatives can be included on the variational problem(Theorem 9). To make the paper self-contained, we begin with a review of the definitions and results from [29]needed to the present work. 2n what follows we consider α, β ∈ ]0 , h ∈ ]0 ,
1[ with h ≪ a, b ∈ R with a < b , and definethe interval I := [ a − h, b + h ]. Definition 1.
Let f : I → R . The h -forward difference operator is defined by ∆ h [ f ]( t ) := f ( t + h ) − f ( t ) h , t ∈ [ a − h, b ] . The h -backward difference operator is defined by ∇ h [ f ]( t ) := f ( t ) − f ( t − h ) h , t ∈ [ a, b + h ] . Remark 1.
Obviously, if f is a differentiable function (in the classical sense), then lim h → ∆ h [ f ]( t ) = lim h → ∇ h [ f ]( t ) = f ′ ( t ) . Definition 2.
The h -scale derivative of f : I → R is defined by (cid:3) h f (cid:3) t ( t ) = 12 (cid:20)(cid:18) ∆ h [ f ]( t ) + ∇ h [ f ]( t ) (cid:19) + i (cid:18) ∆ h [ f ]( t ) − ∇ h [ f ]( t ) (cid:19)(cid:21) , t ∈ [ a, b ] , i = − . (1) For complex valued functions f we define (cid:3) h f (cid:3) t ( t ) = (cid:3) h Re f (cid:3) t ( t ) + i (cid:3) h Im f (cid:3) t ( t ) where Re f and Im f denote, respectively, the real and imaginary part of f . Remark 2.
Again, if f : I → R is a differentiable function, then lim h → (cid:3) h f (cid:3) t ( t ) = f ′ ( t ) . Let C conv ([ a, b ] × ]0 , , C ) be the subspace of C ([ a, b ] × ]0 , , C ) such that for any function g ∈ C conv ([ a, b ] × ]0 , , C ), the limit lim h → g ( t, h )exists for any t ∈ [ a, b ]. Denote by E a complementary space of C conv ([ a, b ] × ]0 , , C ) in C ([ a, b ] × ]0 , , C )and denote by π the projection of C conv ([ a, b ] × ]0 , , C ) ⊕ E onto C conv ([ a, b ] × ]0 , , C ), that is, π : C conv ([ a, b ] × ]0 , , C ) ⊕ E → C conv ([ a, b ] × ]0 , , C ) g := g conv + g E π ( g ) = g conv . In order to define the scale derivative, we need to introduce the following operator: h·i : C ([ a, b ] × ]0 , , C ) → F ([ a, b ] , C ) g
7→ h g i : t lim h → π ( g )( t, h )where F ([ a, b ] , C ) denotes the set of functions f : [ a, b ] → C .Finally, we arrive to the main concept introduced in [29]: the scale derivative of f (withoutthe dependence of h ). Definition 3.
The scale derivative of f ∈ C ( I, C ) , denoted by (cid:3) f (cid:3) t , is defined by (cid:3) f (cid:3) t ( t ) := (cid:28) (cid:3) h f (cid:3) t (cid:29) ( t ) , t ∈ [ a, b ] . (2) Remark 3.
As remarked in [29] , the operator h·i depends on the choice of the complementaryspace E . However, this dependence does not change the form of the properties of this derivative. E is fixed.It is clear that the scale derivative (2) is a linear operator and the scale derivative of a constantis zero. Remark 4.
It is clear that if f : I → R is a differentiable function, then (cid:3) f (cid:3) t ( t ) = f ′ ( t ) . Higher-order scale derivatives can be defined as usual. For a given n ∈ N , denote by I n :=[ a − nh, b + nh ]. The n th scale derivative of a function f : I n → C is the function (cid:3) n f (cid:3) t n definedrecursively by (cid:3) n f (cid:3) t n ( t ) = (cid:3)(cid:3) t (cid:18) (cid:3) n − f (cid:3) t n − (cid:19) ( t ) , t ∈ [ a, b ] , where (cid:3) f (cid:3) t := (cid:3) f (cid:3) t and (cid:3) f (cid:3) t := f .In what follows we will denote C n (cid:3) ([ a, b ] , K ) := { f ∈ C ( I n , K ) | (cid:3) k f (cid:3) t k ∈ C ( I n − k , C ) , k = 1 , , . . . , n } , K = R or K = C . It can be proved that the scale derivative satisfies a scale analogue of Leibniz’s and Barrow’srule in certain subclasses of C ( I, C ).First let us recall the definition of H¨olderian function. Definition 4.
Let f ∈ C ( I, C ) . We say that f is H¨olderian of H¨older exponent α if there existsa constant C > such that, for all s, t ∈ I , the inequality | f ( t ) − f ( s ) | ≤ C | t − s | α holds. The set of H¨olderian functions of H¨older exponent α defined on I is denoted by H α ( I, C ) . Theorem 1 (The scale Leibniz rule [29]) . Let α, β ∈ ]0 , be such that α + β > . For f ∈ H α ( I, R ) and g ∈ H β ( I, R ) , we have (cid:3) ( f.g ) (cid:3) t ( t ) = (cid:3) f (cid:3) t ( t ) .g ( t ) + f ( t ) . (cid:3) g (cid:3) t ( t ) , t ∈ [ a, b ] . Theorem 2 (The scale Barrow rule [29]) . Let f ∈ C (cid:3) ([ a, b ] , R ) be such that lim h → Z ba (cid:18) (cid:3) h f (cid:3) t (cid:19) E ( t ) dt = 0 , (3) where (cid:3) h f (cid:3) t := (cid:16) (cid:3) h f (cid:3) t (cid:17) conv + (cid:16) (cid:3) h f (cid:3) t (cid:17) E . Then, Z ba (cid:3) f (cid:3) t ( t ) dt = f ( b ) − f ( a ) . Remark 5.
Although Theorems 1 and 2 were proven for real valued functions, they still hold forcomplex valued functions (the proofs are similar to the ones given in [29]) . The following result is an easy consequence of the complex versions of Theorems 1 and 2.
Theorem 3 (The scale integration by parts formula) . Let f ∈ H α ( I, C ) and g ∈ H β ( I, C ) besuch that α + β > and lim h → Z ba (cid:18) (cid:3) h ( f · g ) (cid:3) t (cid:19) E ( t ) dt = 0 . Then Z ba (cid:3) f (cid:3) t ( t ) · g ( t ) dt = [ f ( t ) g ( t )] ba − Z ba f ( t ) · (cid:3) g (cid:3) t ( t ) dt. Theorem 4.
Given f ∈ C (cid:3) ([ a, b ] , C ) and t ∈ [ a, b ] such that (cid:3) f (cid:3) t ∈ H α ( I, C ) , lim h → Z ta (cid:18) (cid:3) h f (cid:3) τ (cid:19) E ( τ ) dτ = 0 , and lim h → Z ta (cid:3) h ( (cid:3) f (cid:3) τ · ( t − τ )) (cid:3) τ ! E ( τ ) dτ = 0 , then f ( t ) = f ( a ) + (cid:3) f (cid:3) t ( a )( t − a ) + O ( t − a ) . Proof.
First observe that τ t − τ is of class C and thus (cid:3)(cid:3) τ ( t − τ ) = ddτ ( t − τ ) = − . Using Theorem 2 we conclude that f ( t ) − f ( a ) = Z ta (cid:3) f (cid:3) τ ( τ ) dτ = − Z ta (cid:3) f (cid:3) τ ( τ ) · (cid:3)(cid:3) τ ( t − τ ) dτ. Using the scale integration by parts formula (Theorem 3), we get f ( t ) − f ( a ) = (cid:3) f (cid:3) t ( a )( t − a ) + Z ta (cid:3) f (cid:3) τ ( τ ) · ( t − τ ) dτ = (cid:3) f (cid:3) t ( a )( t − a ) + M ( t − a ) , for some M ∈ C , which ends the proof.We finish this section with the following useful result. Lemma 1.
Let α, β ∈ ]0 , be such that α ≤ β . Then, for all y ∈ H α ( I, C ) and η ∈ H β ( I, C ) , y + η ∈ H α ( I, C ) . The aim is to exhibit several necessary and sufficient conditions in order to determine scale ex-tremals for a certain class of functionals, which domain is the set of H¨olderian functions. Not onlythe Euler-Lagrange equation will be obtained, but also natural boundary conditions will appear.
We consider the following functional: I [ y, T ] = Z Ta L (cid:18) t, y ( t ) , (cid:3) y (cid:3) t ( t ) (cid:19) dt defined on A = { y ∈ H α ( I, R ) : y ( a ) = y a ∧ y ∈ C (cid:3) ([ a, b ] , R ) } , where T ∈ R is such that a ≤ T ≤ b , the Lagrangian L = L ( t, y, v ) : [ a, b ] × R × C → C is of class C , and y a ∈ R is a given fixed real number. We emphasize that we have a free terminal point T and no constraint on y ( T ). Hence, T and y ( T ) become part of the extremal choice process.5 efinition 5. We say that ( y, T ) is a scale extremal of functional I defined on A if, for any η ∈ H β ( I, R ) ∩ C (cid:3) ([ a, b ] , R ) such that η ( a ) = 0 , and δ ∈ R , ddε I [ y + εη, T + εδ ] | ε =0 = 0 . Our first goal is to obtain a necessary and a sufficient condition to ( y, T ) be a scale extremalof the following functional I [ y, T ] = Z Ta L (cid:18) t, y ( t ) , (cid:3) y (cid:3) t ( t ) (cid:19) dty ∈ A , T ∈ [ a, b ] . (P)In the sequel we assume that α + β > α ≤ β . For simplicity of notation, we introducethe operator [ · ] defined by [ y ]( t ) = (cid:18) t, y ( t ) , (cid:3) y (cid:3) t ( t ) (cid:19) . Theorem 5 (The scale Euler–Lagrange equation and natural boundary conditions I) . Let e T ∈ [ a, b ] and e y ∈ A be such that ∂L∂v [ e y ] ∈ H α ( I, C ) and lim h → Z e Ta (cid:18) (cid:3) h (cid:3) t (cid:18) ∂L∂v [ e y ] · η (cid:19) ( t ) (cid:19) E dt = 0 for all η ∈ H β ( I, R ) ∩ C (cid:3) ([ a, b ] , R ) such that η ( a ) = 0 . The pair ( e y, e T ) is a scale extremal offunctional I defined on A if and only if the following conditions hold:1. ∂L∂y [ e y ]( t ) = (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) for all t ∈ [ a, e T ] ;2. ∂L∂v (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = 0 ;3. L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = 0 .Proof. Suppose that ( e y, e T ) is a scale extremal of problem (P). Hence, by definition, ddε I [ e y + εη, e T + εδ ] (cid:12)(cid:12)(cid:12) ε =0 = 0for all η ∈ H β ( I, R ) ∩ C (cid:3) ([ a, b ] , R ) such that η ( a ) = 0, and all δ ∈ R .Note that, by Lemma 1, e y + εη ∈ A . Also note that0 = ddε I [ e y + εη, e T + εδ ] (cid:12)(cid:12)(cid:12) ε =0 = ddε Z e T + εδa L (cid:18) t, e y ( t ) + εη ( t ) , (cid:3) e y (cid:3) t ( t ) + ε (cid:3) η (cid:3) t ( t ) (cid:19) dt ! (cid:12)(cid:12)(cid:12) ε =0 = Z e Ta (cid:20) ∂L∂y [ e y ]( t ) · η ( t ) + ∂L∂v [ e y ]( t ) · (cid:3) η (cid:3) t ( t ) (cid:21) dt + L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) · δ = Z e Ta (cid:20) ∂L∂y [ e y ]( t ) − (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) (cid:21) · η ( t ) dt + (cid:20) ∂L∂v [ e y ]( t ) · η ( t ) (cid:21) e Ta + L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) · δ (4)6ince η ( a ) = 0, then Z e Ta (cid:20) ∂L∂y [ e y ]( t ) − (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) (cid:21) · η ( t ) dt + ∂L∂v [ e y ]( e T ) · η ( e T ) + L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) · δ = 0 (5)If we restrict the variations in (5) to those such that η ( e T ) = 0 and δ = 0, we get Z e Ta (cid:20) ∂L∂y [ e y ]( t ) − (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) (cid:21) · η ( t ) dt = 0 . From the fundamental lemma of the calculus of variations it follows that ∂L∂y [ e y ]( t ) − (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) = 0 (6)for all t ∈ [ a, e T ].If we restrict in (5) to those η such that η ( e T ) = 0, we get Z e Ta (cid:20) ∂L∂y [ e y ]( t ) − (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) (cid:21) · η ( t ) dt + L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) · δ = 0 . (7)Substituting the scale Euler-Lagrange equation (6) into (7) we obtain L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) · δ = 0 . By the arbitrariness of δ we get L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = 0 . Substituting δ = 0 and the scale Euler-Lagrange equation (6) into (5), we have that ∂L∂v [ e y ]( e T ) · η ( e T ) = 0 . From the arbitrariness of η , we conclude that ∂L∂v (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = 0 . Corollary 1. ([29])
Suppose that T is fixed in problem (P) and the set of admissible functions isgiven by B := A ∩ { y ∈ H α ( I, R ) : y ( T ) = y T } for some fixed real number y T . Let e y ∈ B be such that ∂L∂v [ e y ] ∈ H α ( I, C ) and lim h → Z Ta (cid:18) (cid:3) h (cid:3) t (cid:18) ∂L∂v [ e y ] · η (cid:19) ( t ) (cid:19) E dt = 0 for all η ∈ H β ( I, R ) ∩ C (cid:3) ([ a, b ] , R ) such that η ( a ) = η ( T ) = 0 . Then e y is a scale extremal offunctional I in the class B if and only if ∂L∂y [ e y ]( t ) = (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) , ∀ t ∈ [ a, T ] . emark 6. (cf. [33]) If we restrict the set of admissible functions to be the set { y ∈ C ([ a, b ] , R ) : y ( a ) = y a } , then problem (P) reduces to the classical variational problem I [ y, T ] = Z Ta L ( t, y ( t ) , y ′ ( t )) dt → extremizey ∈ C ([ a, b ] , R ) y ( a ) = y a , (8) and, by Theorem 5, we can conclude that if ( e y, e T ) is an extremizer (that is, minimizer or maxi-mizer) of problem (8) , then1. ∂L∂y [ e y ]( t ) = ddt (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) for all t ∈ [ a, e T ] ;2. ∂L∂v (cid:16) e T , e y ( e T ) , e y ′ ( e T ) (cid:17) = 0 ;3. L (cid:16) e T , e y ( e T ) , e y ′ ( e T ) (cid:17) = 0 . Doing similar calculations as done in the proof of Theorem 5 one can prove the following result.
Theorem 6 (The scale Euler–Lagrange equation and natural boundary conditions II) . Let e T ∈ [ a, b ] and e y ∈ C (cid:3) ([ a, b ] , R ) be such that ∂L∂v [ e y ] ∈ H α ( I, C ) and lim h → Z e Ta (cid:18) (cid:3) h (cid:3) t (cid:18) ∂L∂v [ e y ] · η (cid:19) ( t ) (cid:19) E dt = 0 for all η ∈ H β ( I, R ) ∩ C (cid:3) ([ a, b ] , R ) . The pair ( e y, e T ) is a scale extremal of functional I defined on C (cid:3) ([ a, b ] , R ) if and only if the following conditions hold:1. ∂L∂y [ e y ]( t ) = (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) for all t ∈ [ a, e T ] ;2. ∂L∂v (cid:18) a, e y ( a ) , (cid:3) e y (cid:3) t ( a ) (cid:19) = 0 ;3. ∂L∂v (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = 0 ;4. L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = 0 . Corollary 2. ([32])
Suppose that T is fixed in problem (P) and that the boundary conditions y ( a ) = y a and y ( T ) = y T are not present. Let D = { y ∈ H α ( I, R ) : y ∈ C (cid:3) ([ a, b ] , C ) } , and e y ∈ D be such that ∂L∂v [ e y ] ∈ H α ( I, C ) , and lim h → Z Ta (cid:18) (cid:3) h (cid:3) t (cid:18) ∂L∂v [ e y ] · η (cid:19) ( t ) (cid:19) E dt = 0 for all η ∈ H β ( I, R ) ∩ C (cid:3) ([ a, b ] , R ) . Then e y is a scale extremal of functional I in the class D ifand only if the following conditions hold: . ∂L∂y [ e y ]( t ) = (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) for all t ∈ [ a, T ] ;2. ∂L∂v (cid:18) a, e y ( a ) , (cid:3) e y (cid:3) t ( a ) (cid:19) = 0 ;3. ∂L∂v (cid:18) T, e y ( T ) , (cid:3) e y (cid:3) t ( T ) (cid:19) = 0 . In the proof of Theorem 5 we proved that ( e y, e T ) is a scale extremal of problem (P), if and onlyif, for arbitrary η ∈ H β ( I, R ) ∩ C (cid:3) ([ a, b ] , R ) such that η ( a ) = 0, and δ ∈ R , Z e Ta (cid:20) ∂L∂y [ e y ]( t ) − (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) (cid:21) · η ( t ) dt + ∂L∂v [ e y ]( e T ) · η ( e T ) + L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) · δ = 0 . (9)In what follows we will write η ( e T ) in terms of δ , the increment over time, and the incrementover space, δ e y e T := ( e y + η )( e T + δ ) − e y ( e T ) . Assuming that e y, η ∈ C (cid:3) ([ a, b ] , R ) and for those η such that (cid:3) η (cid:3) t ( e T ) = 0, and | δ | ≪
1, then, byTheorem 4, we deduce that( e y + η )( e T + δ ) − ( e y + η )( e T ) = (cid:3) e y (cid:3) t ( e T ) · δ + O ( δ )and so we obtain the formula for the increment over space δ e y e T = (cid:3) e y (cid:3) t ( e T ) · δ + η ( e T ) + O ( δ )which is equivalent to η ( e T ) = δ e y e T − (cid:3) e y (cid:3) t ( e T ) · δ + O ( δ ) . (10)Substituting (6) and (10) into (9) we get δ (cid:20) L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) − ∂L∂v [ e y ]( e T ) · (cid:3) e y (cid:3) t ( e T ) (cid:21) + δ e y e T · ∂L∂v [ e y ]( e T ) + O ( δ ) = 0 . (11)Depending on the constraints that may be imposed over the terminal point T and/or over theboundary condition y ( T ), several natural boundary conditions can be obtained from condition(11). Obviously, if they are both fixed we do not get any extra condition (see [29]).Next we consider the case where the boundary conditions y ( a ) and y ( T ) are fixed and T isfree. Theorem 7. [The scale Euler–Lagrange equation and natural boundary conditions III] In theconditions of Theorem 5, the pair ( e y, e T ) is a scale extremal of functional I defined on { y ∈ H α ( I , R ) : y ( a ) = y a ∧ y ( T ) = y T ∧ y ∈ C (cid:3) ([ a, b ] , R ) } , where y a , y T ∈ R are fixed real numbers, if and only if the following conditions hold:1. ∂L∂y [ e y ]( t ) = (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) for all t ∈ [ a, e T ] ;2. L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = ∂L∂v [ e y ]( e T ) · (cid:3) e y (cid:3) t ( e T ) . roof. In this case, δ e y e T = 0 and δ is arbitrary. Thus from (11) we obtain the condition L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = ∂L∂v [ e y ]( e T ) · (cid:3) e y (cid:3) t ( e T ) . Now we shall generalize Theorem 7 considering the case where we have the boundary condition e y ( T ) = ψ ( T ), where ψ is a given function of class C (cid:3) ([ a, b ] , R ), and T is free. Theorem 8 (The scale Euler–Lagrange equation and natural boundary conditions IV) . Let ψ bea given function of class C (cid:3) ([ a, b ] , R ) . In the conditions of Theorem 5, the pair ( e y, e T ) is a scaleextremal of functional I defined on { y ∈ H α ( I , R ) : y ( a ) = y a ∧ y ( T ) = ψ ( T ) ∧ y ∈ C (cid:3) ([ a, b ] , R ) } if and only if the following conditions hold:1. ∂L∂y [ e y ]( t ) = (cid:3)(cid:3) t (cid:18) ∂L∂v [ e y ] (cid:19) ( t ) for all t ∈ [ a, e T ] ;2. L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = ∂L∂v [ e y ]( e T ) · (cid:18) (cid:3) e y (cid:3) t ( e T ) − (cid:3) ψ (cid:3) t ( e T ) (cid:19) . Proof.
From Theorem 4 we can conclude that δ e y e T = ψ ( e T + δ ) − ψ ( e T )= (cid:3) ψ (cid:3) t ( e T ) · δ + O ( δ ) . (12)Replacing (12) into (11), the arbitrariness of δ allows us to deduce the condition L (cid:18) e T , e y ( e T ) , (cid:3) e y (cid:3) t ( e T ) (cid:19) = ∂L∂v [ e y ]( e T ) · (cid:18) (cid:3) e y (cid:3) t ( e T ) − (cid:3) ψ (cid:3) t ( e T ) (cid:19) . In this section we consider the following higher-order functional JJ [ y, T ] = Z Ta L (cid:18) t, y ( t ) , (cid:3) y (cid:3) t ( t ) , · · · , (cid:3) n y (cid:3) t n ( t ) (cid:19) dt (13)defined in the class E = { y ∈ H α ( I n , R ) : y ( a ) = y a , (cid:3) y (cid:3) t ( a ) = y a , · · · , (cid:3) n − y (cid:3) t n − ( a ) = y n − a ∧ y ∈ C n (cid:3) ([ a, b ] , R ) } , where the Lagrangian L = L ( t, y, v , v , · · · , v n ) : [ a, b ] × R × C n → C is of class C , T ∈ R is suchthat a ≤ T ≤ b , and y a ∈ R , y a , · · · , y n − a ∈ C are given fixed numbers. Definition 6.
We say that ( y, T ) is a scale extremal of functional J defined on E if, for any η ∈ H β ( I n , R ) ∩ C n (cid:3) ([ a, b ] , R ) such that η ( a ) = (cid:3) η (cid:3) t ( a ) = · · · = (cid:3) n − η (cid:3) t n − ( a ) = 0 , and δ ∈ R , ddε J [ y + εη, T + εδ ] | ε =0 = 0 .
10n order to simplify notations we will denote[ y ] n ( t ) := (cid:18) t, y ( t ) , (cid:3) y (cid:3) t ( t ) , · · · , (cid:3) n y (cid:3) t n ( t ) (cid:19) . Theorem 9 (The higher-order scale Euler–Lagrange equation and natural boundary conditions) . Let e T ∈ [ a, b ] and e y ∈ E be such that1. ∂L∂v i [ e y ] n ∈ H α ( I n , C ) for all i = 1 , , · · · , n ,2. for all i = 1 , , · · · , n and k = 0 , , · · · , i − , lim h → Z e Ta (cid:18) (cid:3) h (cid:3) t (cid:18) (cid:3) k (cid:3) t k ( ∂L∂v i [ e y ] n ) · (cid:3) i − k − η (cid:3) t i − k − (cid:19) ( t ) (cid:19) E dt = 0 , for all η ∈ H β ( I n , R ) ∩ C n (cid:3) ([ a, b ] , R ) such that η ( a ) = 0 , (cid:3) η (cid:3) t ( a ) = 0 , · · · , (cid:3) n − η (cid:3) t n − ( a ) = 0 .The pair ( e y, e T ) is a scale extremal of functional J on the class E if and only if the followingconditions hold:1. ∂L∂y [ e y ] n ( t ) + n X i =1 ( − i (cid:3) i (cid:3) t i (cid:18) ∂L∂v i (cid:19) [ e y ] n ( t ) = 0 for all t ∈ [ a, e T ] ;2. n X k = i ( − k − i (cid:3) k − i (cid:3) t k − i (cid:18) ∂L∂v k (cid:19) [ e y ] n ( e T ) = 0 , ∀ i = 1 , , · · · , n ; L [ e y ] n ( e T ) = 0 .Proof. Suppose that ( e y, e T ) is a scale extremal of functional J on the class E . For variationfunctions, we consider ( e y + εη, e T + εδ ), with η be such that η ( a ) = (cid:3) η (cid:3) t ( a ) = · · · = (cid:3) n − η (cid:3) t n − ( a ) = 0 . Using the definition of scale extremal, we conclude that Z e Ta ∂L∂y [ e y ] n ( t ) · η ( t ) + n X i =1 ∂L∂v i [ e y ] n ( t ) · (cid:3) i η (cid:3) t i ( t ) ! dt + L [ e y ] n ( e T ) · δ = 0 . Applying the integration by parts formula, we get Z e Ta ∂L∂y [ e y ] n ( t ) + n X i =1 ( − i (cid:3) i (cid:3) t i ( ∂L∂v i )[ e y ] n ( t ) ! · η ( t ) dt + n X i =1 ∂L∂v i [ e y ] n ( e T ) · (cid:3) i − η (cid:3) t i − ( e T ) + i − X k =1 ( − k (cid:3) k (cid:3) t k ( ∂L∂v i )[ e y ] n ( e T ) · (cid:3) i − − k η (cid:3) t i − − k ( e T ) ! + L [ e y ] n ( e T ) · δ = 0 . Considering δ = 0 and η ( e T ) = (cid:3) η (cid:3) t ( e T ) = · · · = (cid:3) n − η (cid:3) t n − ( e T ) = 0 we obtain the higher-order scaleEuler-Lagrange equation. Similarly as done in the proof of Theorem 5, for appropriate variations,we obtain the pretended natural boundary conditions.Clearly, all the results presented in Subsection 3.1 can be generalized for higher-order varia-tional problems. 11 Conclusions
In the present paper we study variational problems when the dynamic of the trajectories arenondifferentiable. To overcome this situation, we considered the scale derivative as presented byCresson and Greff in [29] and [30], which has shown some applications in physics, e.g., trajectoriesof quantum mechanics, fractals and scale-relativity theory. The main aim was to find necessary andsufficient conditions that a pair ( y, T ) must satisfy in order to be an extremal of a given functional,where y is the trajectory and T the end-time of the integral. We considered the existence or notof boundary conditions on the initial and end time, as well with higher-order scale derivatives. Acknowledgments
This work was supported by Portuguese funds through the CIDMA - Center for Research and De-velopment in Mathematics and Applications, and the Portuguese Foundation for Science and Tech-nology (FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia), within project PEst-OE/MAT/UI4106/2014.