Published online in International Academic Press (www.IAPress.org)
General quantum variational calculus
Artur M. C. Brito da Cruz
1,2,∗, Nat´alia Martins
21Escola Superior de Tecnologia de Set´ubal, Campus do IPS, Estefanilha, 2914-508 Set´ubal, Portugal 2Center for Research and Development in Mathematics and Applications (CIDMA),
Department of Mathematics, University of Aveiro, Portugal
Abstract We develop a new variational calculus based in the general quantum difference operator recently introduced by Hamza et al. In particular, we obtain optimality conditions for generalized variational problems where the Lagrangian may depend on the endpoints conditions and a real parameter, for the basic and isoperimetric problems, with and without fixed boundary conditions. Our results provide a generalization to previous results obtained for theq- and Hahn-calculus.
Keywords General quantum calculus; Hahn’s difference operator; Jackson’s integral; quantum calculus; calculus of variations; Euler–Lagrange equation; generalized natural boundary conditions; isoperimetric problem.
AMS 2010 subject classifications39A13, 39A70, 49J05, 49K05, 49K15. DOI:10.19139/soic.v6i1.467
1. Introduction
It is well-known that many problems in physics, economics, mechanics and calculus of variations can be formulated using integral functionals and that many real life phenomena are characterized by functions that are not differentiable in the classical sense. In the literature there are different approaches to deal with the non-differentiability of the admissible functions in problems involving integral functionals. In this paper, we are concerned with the quantum approach.
The word quantum is usually referred to the smallest discrete quantity of some physical property and it comes from the Latin word ”quantus”, which literally means how many. The quantum calculus is usually referred in mathematics as the calculus without limits and it replaces the classical derivative by a difference operator. One type of quantum differential operator is the Hahn quantum derivative introduced in 1949 by Hahn [13]. This operator has been a useful tool in the construction of ortogonal polynomials and in approximation problems [5,12,18,23]. The Hahn calculus has its begining only in 2009 with the Ph.D. thesis of Aldwoah [1], where the author solved an open problem with 60 years: the construction of the inverse operator of Hahn’s derivative and the development of the calculus associated with this operator. One year later, the Hahn variational calculus started with the publication of Malinowska and Torres [20]. Due to its importance, new results in Hahn quantum variational calculus have been published in the literature [8,19]; we also recommend the book [21] for more details on the topic. We refer the interested reader to [2,3,4,7,10,22] for details in calculus of variations based in different quantum operators.
∗Correspondence to: Artur M. C. Brito da Cruz (Email: artur.cruz@estsetubal.ips.pt), Escola Superior de Tecnologia de Set´ubal, Campus
do IPS, Estefanilha, 2914-508 Set´ubal, Portugal.
Recently, Hamza et al. [14] defined the general quantum difference operator by Dβ[f ] (t) = f (β (t))− f (t) β (t)− t , t̸= β (t) , f′(t) , t = β (t) ,
where β : I → I is a strictly increasing continuous function with a unique fixed point s0∈ I, where I is a subinterval ofR, and such that(t− s0)(β(t)− t)60for allt∈ I, and developed the general quantum calculus which generalizes the well-knownq-quantum calculus (also known in the literature as Jackson’s calculus) [16] and Hanh’s quantum calculus [9,13].
The main purpose of this paper is to study different types of variational problems involving the general quantum difference operator: the basic problem and the isoperimetric problem, with and without fixed boundary conditions, for problems of the calculus of variations with a Lagrangian that may depend on the endpoints conditions and a real parameter.
The structure of the paper is as follows. In Section 2 we present the necessary definitions and results for the general quantum calculus. Our results are given in Section 3: the general quantum Euler-Lagrange equation for the basic problem (Theorem11), the transversality conditions for the basic problem with free boundary conditions (Theorem15), necessary optimality conditions for the isoperimetric problem (Theorem18and Theorem19) and a sufficient optimality condition (Theorem14). To finalize the paper we present some illustrative examples and some concluding remarks.
2. Preliminaries
In this section we start with the definition of the general quantum derivative. LetI⊆R be an interval,f : I→R be
a real function andβ : I → Ibe a strictly increasing and continuous function that has a unique fixed points0∈ I that satisfies the following inequality:
(t− s0) (β (t)− t)60 for allt∈ I, (1) where the equality holds only ift = s0. Hamza et al. [14] defined the general quantum difference operator by
Dβ[f ] (t) = f (β (t))− f (t) β (t)− t , t̸= s0 f′(s0) , t = s0
provided thatf′(s0)exists (and is finite) in the usual sense. One says thatf isβ- differentiable onIiff′(s0)exists and we callDβ[f ] (t)theβ−derivative off att.
We remark that the general quantum difference operator Dβ can also be defined for strictly increasing and continuous functionsβthat have a unique fixed points0and satisfy(t− s0) (β (t)− t)>0, for allt∈ I(see [14]). It is clear that every choice of the functionβgives a new difference operator and that this operator generalize the well-known Jackson and Hahn operators: ifβ(t) = qt, for someq∈]0, 1[, then we obtain the Jacksonq-difference operator [16]; if β(t) = qt + ω, for some q∈]0, 1[ and ω>0, then we obtain the Hahn’s quantum difference operator [13]. For a general introduction to the q-calculus and the Hahn’s calculus, we refer the reader to the book [17] and to the Ph.D. thesis [1], respectively.
In the following, we present the notions and results from the general quantum difference calculus that are needed in this paper. We proceed with some basic properties of the general quantum difference operator.
Theorem 1 ([14])
Letf andgbeβ-differentiable onIandk∈R. One has:
2. Dβ[kf ] (t) = kDβ[f ] (t) ; 3. Dβ[f g] (t) = Dβ[f ] (t) g (t) + f (β (t)) Dβ[g] (t) = Dβ[f ] (t) g (β (t)) + f (t) Dβ[g] (t) ; 4. Dβ [ f g ] (t) = Dβ[f ] (t) g (t)− f (t) Dβ[g] (t) g (t) g (β (t)) , providedg (t) g (β (t))̸= 0; 5. f (β (t)) = f (t) + (β (t)− t) Dβ[f ] (t) ;
6. IfDβ[f ] (t) = 0, for allt∈ I, thenf (t) = f (s0),t∈ I.
It is clear that iff is β-differentiable, thenf is continuous ats0. We note here that there exist discontinuous functions that areβ-differentiable (for an example see [14]), something that cannot happen in the classical theory and can be seen as an advantage of the general quantum calculus.
In what follows, let us denote
βk(t) = β◦ β ◦ . . . ◦ β
| {z }
ktimes
(t) and β0(t) = t, t∈ I.
The following lemma is a useful property of functionβ.
Lemma 1 ([14])
The sequence of functions{βk(t)}
k∈N0converges uniformly to the fixed points0on every compact intervalJ ⊆ I containings0.
Now we present the definition ofβ−integral.
Definition 2 ([14])
Leta, b∈ Ianda < b. Forf : I→R theβ-integral offfromatobis given by ∫ b a f (t) dβt = ∫ b s0 f (t) dβt− ∫ a s0 f (t) dβt where ∫ x s0 f (t) dβt = +∞ ∑ k=0 ( βk(x)− βk+1(x))f(βk(x)),x∈ I
provided that the series converges atx = aandx = b.f is calledβ−integrable onIif the series converge ataand
b, for alla, b∈ I.
The following result is also very useful.
Theorem 3 ([14])
Iff →R is continuous ats0∈ I, thenf isβ-integrable onI.
Remark 1
Theβ-integral generalizes the Jacksonq-integral and the Hahn integral. Whenβ (t) = qt forq∈ ]0, 1[, we obtain the Jacksonq-integral [17]:
∫ b a f (t) dqt = ∫ b 0 f (t) dqt− ∫ a 0 f (t) dqt where ∫ x 0 f (t) dqt = x (1− q) +∞ ∑ k=0 qkf(xqk),x∈ I.
Whenβ (t) = qt + ω, forq∈ ]0, 1[andω>0, we obtain the Hahn integral [1]: ∫ b a f (t) dq,ωt = ∫ b ω0 f (t) dq,ωt− ∫ a ω0 f (t) dq,ωt
where ∫ x ω0 f (t) dq,ωt = (x (1− q) − ω) +∞ ∑ k=0 qkf(xqk+ ω [k]q),x∈ I, and whereω0= ω 1− q and[k]q = 1− qk 1− q .
Theβ-integral has the following properties:
Theorem 4 ([14])
Letf, g : I →R beβ−integrable onI,a, b, c∈ Iandk∈R. Then,
1. ∫ b a (f + g) (t) dβt = ∫ b a f (t) dβt + ∫ b a g (t) dβt; 2. ∫ b a (kf ) (t) dβt = k ∫ b a f (t) dβt; 3. ∫ a a f (t) dβt = 0; 4. ∫ b a f (t) dβt =− ∫ a b f (t) dβt; 5. ∫ b a f (t) dβt = ∫ c a f (t) dβt + ∫ b c f (t) dβt. In what follows, for a givens∈ I, we denote
[s]β :={βk(s) : k∈N0 }
∪ {s0}. and fors0∈ [a, b] ⊆ I,
[a, b]β:= [a]β∪ [b]β; [a, b]βis called theβ-interval with extreme pointsaandb.
Theorem 5 ([14])
Letf, g : I →R be β-integrable functions on I and let a, b∈ I such that a < s0< b. If |f (t)|6g (t) for all
t∈ [a, b]β, then 1. ∫ b s0 f (t) dβt 6 ∫ b s0 g (t) dβt; 2. ∫ a s0 f (t) dβt 6− ∫ a s0 g (t) dβt; 3. ∫ b a f (t) dβt 6 ∫ b a g (t) dβt.
Consequently, if g(t)>0, for allt∈ [a, b]β, then the inequalities ∫ b s0 g(t)dβt>0and ∫ b a g(t)dβt>0 hold. Remark 2 We remark that:
2. It may happens thatg>0and ∫ b
a
g (t) dβt < 0. For example, for the functiong : I→R defined by
g (t) = λ1 for t∈ [a]β 0 otherwise
for some fixedλ1∈R+anda > s0, we have ∫ b a g (t) dβt = +∞ ∑ k=0 ( βk(b)− βk+1(b))g(βk(b))− +∞ ∑ k=0 (
βk(a)− βk+1(a))g(βk(a)) = −λ1(a− s0) < 0.
Hamza et al. also have proven the Fundamental Theorem ofβ-calculus.
Theorem 6 (Fundamental Theorem ofβ−calculus [14])
Letf : I→R be continuous ats0. LetF : I→R be the function defined by
F (x) =
∫ x s0
f (t) dβt.
Then,F is continuous ats0. Furthermore,Dβ[F ] (x)exists for everyx∈ Iand
Dβ[F ] (x) = f (x) . Conversely, iff isβ-differentiable onI, then
∫ b a
Dβ[f ] (t) dβt = f (b)− f (a) for alla, b∈ I.
Next, we present theβ-integration by parts which is essential to prove our main results.
Theorem 7 (β−integration by parts [14])
Iff, g : I →R areβ-differentiable andDβ[f ]andDβ[g]are continuous ats0, then ∫ b a f (t) Dβ[g] (t) dβt = f (t) g (t) b a− ∫ b a Dβ[f ] (t) g (β (t)) dβt, a, b∈ I. The next result is a main tool to obtain necessary optimality conditions for our variational problems.
Lemma 2 (Fundamental lemma of the general quantum variational calculus [14]) Letf : [a, b]β→R be continuous ats0. Then,
∫ b a
f (t) η (β (t)) dβt = 0
for all functionsη : [a, b]β→R continuous ats0withη(a) = η(b) = 0if and only iff (t) = 0for allt∈ [a, b]β. The following definition and lemma are important for the proofs of our results.
Definition 8
Lets∈ Iandg : [s]β× ]
−¯θ, ¯θ[→R. We say thatg (t,·)is differentiable atθ0uniformly in[s]βif for everyε > 0 there existsδ > 0such that
0 <|θ − θ0| < δ ⇒ g (t, θ)− g (t, θ0) θ− θ0 − ∂ 2g (t, θ0) < ε
Lemma 3 ([14])
Let s∈ I. Assume that g : [s]β× ]
−¯θ, ¯θ[→R is differentiable at θ0 uniformly in [s]β and that G (θ) = ∫ s
s0
g (t, θ) dβt, in a neighborhood ofθ0, and ∫ s
s0
∂2g (t, θ0) dβtexist. Then,Gis differentiable atθ0with
G′(θ0) = ∫ s
s0
∂2g (t, θ0) dβt.
3. Main results
The main objective of this section is to introduce the general quantum variational calculus. For this purpose we consider the following generalized variational problem:
L [y, ζ] =
∫ b a
L (t, y (β (t)) , Dβ[y] (t) , y(a), y(b), ζ) dβt−→ extremize (2) where by extremize we mean minimize or maximize,ζis a real parameter, andy∈ Y, where
Y :={y : I→R| yandDβ[y] are bounded on [a, b]β and continuous ats0 } endowed with the norm
∥y∥ = sup
t∈[a,b]β
|y (t)| + sup
t∈[a,b]β
|Dβ[y] (t)| .
We will consider problem (2) with or without fixed boundary conditions and also with an isoperimetric restriction. The Lagrangian functionLis assumed to satisfy the following hypotheses:
(H1) (u0, u1, u2, u3, u4)−→ L (t, u0, u1, u2, u3, u4)is aC1 (
R5,R)function for anyt∈ I; (H2) t−→ L (t, y (β(t)) , Dβ[y] (t) , y(a), y(b), ζ)is continuous ats0for anyy∈ Y;
(H3) functionst→ ∂i+2L (t, y (β(t)) , Dβ[y] (t) , y(a), y(b), ζ),i = 0, 1, . . . , 4belong toY, for ally∈ Y; where∂iLdenotes the partial derivative ofLwith respect to itsith-coordinate.
The motivation to study generalized variational problems where the Lagrangian depends on the state values can be found in economics problems ([24]); the dependence of a real parameter is important, for example, in physical problems ([15]).
In Section 3.1we derive optimality conditions for problem (2) in the class of functionsy∈ Y satisfying the boundary conditions:
y (a) = αandy (b) = γ (3)
for some fixed α, γ∈R. Then we consider problem (2) without fixed boundary conditions. In this case, two additional necessary conditions, usually called natural boundary conditions or transversality conditions, are obtained. Finally, in Section 3.2 we study problem (2) in the class of functions y∈ Y satisfying the integral constraint
I [y, ζ] =
∫ b a
F (t, y (β (t)) , Dβ[y] (t) , y(a), y(b), ζ) dβt = τ (4) for someτ∈R.
3.1. Generalized quantum variational problems
We begin this section with some basic definitions that are useful in what follows.
Definition 9
We say thatyis an admissible function for problem (2) ify∈ Y;y∈ Yis said an admissible function for problem (2)-(3) if y satisfies the boundary conditions (3). We say that a function η∈ Y is an admissibile variation for problem (2)-(3) ifη (a) = η (b) = 0.
Definition 10
We say that (y∗, ζ∗) is a local minimimizer (resp. local maximizer) for problem (2) ((and (2)-(3)) if y∗ is an admissible function and there existsγ > 0such that
L [y∗, ζ∗]6L [y, ζ] (resp. L [y∗, ζ∗]>L [y, ζ])
for admissible functionyandζ∈R with∥y∗− y∥ < γand|ζ − ζ∗| < γ. In what follows and in order to simplify expressions, we introduce the notation
[y; ζ]β(t) := (t, y (β (t)) , Dβ[y] (t) , y(a), y(b), ζ) . Therefore,
L [y, ζ] =
∫ b a
L [y; ζ]β(t) dβt.
For an admissible variationη, an admissible functiony, a real parameterζ, and a realδ, we define the function
ϕ : ]−¯ϵ, ¯ϵ[ →R by
ϕ (ϵ) =L [y + ϵη, ζ + ϵδ] .
The following result is an immediate consequence of Lemma3.
Lemma 4
For an admissible variationη, an admissible functiony, andδ∈R, let g (t, ϵ) = L [y + ϵη; ζ + ϵδ]β(t)
forϵ∈ ]−¯ϵ, ¯ϵ[. Assume that:
(i) g (t,·)is differentiable at0uniformly in[a, b]β; (ii) ∫ a s0 L [y + ϵη; ζ + ϵδ]β(t)dβtand ∫ b s0
L [y + ϵη; ζ + ϵδ]β(t)dβtexist forϵin a neighborhood of0; (iii) ∫ a s0 ∂2g (t, 0) dβtand ∫ b s0 ∂2g (t, 0) dβtexist. Then, ϕ′(0) = ∫ b a ( ∂2L [y; ζ]β(t)· η (β (t)) + ∂3L [y; ζ]β(t)· Dβ[η] (t)
+∂4L [y; ζ]β(t)· η(a) + ∂5L [y; ζ]β(t)· η(b) + ∂6L [y; ζ]β· δ )
dβt.
We can now prove a necessary condition for an admissible pair(y∗, ζ∗)to be a local extremizer of problem (2)-(3).
Theorem 11 (General quantum Euler-Lagrange equation)
Under hypotheses (H1)-(H3) and (i)-(iii) of Lemma4on the Lagrangian functionL, if the pair(y∗, ζ∗)is a local extremizer for problem (2)-(3), then(y∗, ζ∗)satisfies theβ-Euler-Lagrange equation
Dβ[∂3L] [y; ζ]β(t) = ∂2L [y; ζ]β(t) , t∈ [a, b]β (5)
and ∫
b
a
Proof
Let(y∗, ζ∗)be a local extremizer for problem (2)-(3),ηan admissible variation andδ∈R. A necessary condition
for(y∗, ζ∗)to be an extremizer is given byϕ′(0) = 0. Therefore, by Lemma4, ∫ b a ( ∂2L [y∗; ζ∗]β(t)· η (β (t)) + ∂3L [y∗; ζ∗]β(t)· Dβ[η] (t) +∂4L [y∗; ζ∗]β(t)· η(a) + ∂5L [y∗; ζ∗]β(t)· η(b) + ∂6L [y∗; ζ∗]β· δ ) dβt = 0. Usingβ-integration by parts and observing thatη(a) = η(b) = 0, we obtain
∫ b a ∂3L [y∗; ζ∗]β(t)· Dβ[η] (t) dβt = ∂3L [y∗; ζ∗]β(t)· η (t) b a− ∫ b a Dβ[∂3L] [y∗; ζ∗]β(t)· η (β (t)) dβt =− ∫ b a Dβ[∂3L] [y∗; ζ∗]β(t)· η (β (t)) dβt and therefore ∫ b a ( ∂2L [y∗; ζ∗]β(t)− Dβ[∂3L] [y∗; ζ∗]β(t) ) η (β (t)) dβt + ∫ b a ∂6L [y∗; ζ∗]β(t)· δ dβt = 0. Takingδ = 0and using the arbitrariness ofη, we can conclude by Lemma2, that
∂2L [y∗; ζ∗]β(t)− Dβ[∂3L] [y∗; ζ∗]β(t) = 0, for allt∈ [a, b]β. Taking the admissible variation to be null we obtain
∫ b a
∂6L [y∗; ζ∗]β(t)· δ dβt = 0 and, by the arbitrariness ofδ, we get ∫
b
a
∂6L [y∗; ζ∗]β(t) dβt = 0, as intended.
In the particular case where the Lagrangian functionLdoes not depend on the state valuesy(a)andy(b), and on the real parameterζ, we obtain the following necessary optimality condition for the simplest variational problem, where the Lagrangian L satisfies conditions (H1)-(H3) and (i)-(iii) of Lemma 4, adapted to the case where L
depends only on the arguments(t, y(β(t)), Dβ[y] (t)).
Corollary 1
Ify∗is a local extremizer to problem
L [y] =
∫ b a
L (t, y (β (t)) , Dβ[y] (t)) dβt−→ extremize satisfying the boundary conditions (3), theny∗satisfies the Euler-Lagrange equation
Dβ[∂3L] (t, y (β (t)) , Dβ[y] (t)) = ∂2L (t, y (β (t)) , Dβ[y] (t)) (7) for allt∈ [a, b]β.
Remark 3
If we specify the functionβand consider the simplest problem of calculus of variations, we obtain the following necessary optimality conditions, proven in [6] and [20], respectively:
1. ifβ (t) = qt, forq∈ ]0, 1[, then we obtain theq-Euler-Lagrange equation
∂2L(t, y (qt) , Dq[y](t)) = Dq[∂3L](t, y (qt)) , Dq[y](t));
2. ifβ (t) = qt + ω, forq∈ ]0, 1[andω>0, then we obtain the Hahn’s quantum Euler-Lagrange equation
∂2L(t, y (qt + ω) , Dq,ω[y](t)) = Dq,ω[∂3L](t, y (qt + ω) , Dq,ω[y](t)).
Definition 12
We say that a pair(y, ζ)is an extremal to problem (2)-(3) if(y, ζ)satisfies theβ-Euler-Lagrange equation (5) and condition (6).
In order to prove a sufficient optimality condition for problem (2)-(3), that is useful in the search of extremizers, we introduce the following definition.
Definition 13
Given a function L : I×R5→R, we say that L (t, u
0, . . . , u4) is jointly convex (resp. jointly concave) in
(u0, . . . , u4)if and only if∂iL,i = 2, . . . , 6, are continuous and satisfy the following condition:
L (t, u0+ u0, . . . , u4+ u4)− L (t, u0, . . . , u4) > (resp.6) 6 ∑ i=2 ∂iL (t, u0, . . . , u4) ui−2
for all(t, u0, . . . , u4),(t, u0+ u0, . . . , u4+ u4)∈ [a, b]β×R5.
Theorem 14 (Sufficient optimality condition)
Suppose that a6s0< b or a < s06b and let L (t, u0, . . . , u4) be jointly convex (resp. jointly concave) in
(u0, . . . , u4). If (y∗, ζ∗)is an extremal, then(y∗, ζ∗)is a global minimizer (resp. global maximizer) to problem (2)-(3).
Proof
Without loss of generality, let us consider thatLis jointly convex in(u0, . . . , u4). For any admissible variationη andδ∈R, it follows that
L [y∗+ η, ζ∗+ δ]− L [y∗, ζ∗] = ∫ b a ( L [y∗+ η; ζ∗+ δ]β(t)− L [y∗; ζ∗]β(t))dβt >∫ b a ( ∂2L [y∗; ζ∗]β(t)· η (β(t)) + ∂3L [y∗; ζ∗]β(t)· Dβ[η] (t) + ∂4L [y∗; ζ∗]β(t)· η(a) + ∂5L [y∗; ζ∗]β(t)· η(b) + ∂6L [y∗; ζ∗]β(t)· δ ) dβt.
Using integration by parts, the fact thatη(a) = η(b) = 0and observing that(y∗, ζ∗)satisfies theβ-Euler-Lagrange equation and condition (6), we get
L [y∗, ζ∗]6L [y∗+ η, ζ∗+ δ]
as intended.
Remark 4
We emphasize that conditiona6s0< bora < s06bis mandatory in Theorem14in order to apply Theorem5. Another important question is to find optimality conditions that extremals has to satisfy in the case we have no boundary constraints on the set of admissible functions.
Theorem 15 (Natural boundary conditions)
Under hypotheses (H1)-(H3) and (i)-(iii) of Lemma4on the Lagrangian functionL, if the pair(y∗, ζ∗)is a local extremizer for problem (2), then(y∗, ζ∗)satisfies theβ-Euler-Lagrange equation (5) and condition (6). Moreover,
1. ify (a)is free, then the natural boundary condition
∂3L [y∗; ζ∗]β(a) = ∫ b
a
∂4L [y∗; ζ∗]β(t) dβt holds;
2. ify (b)is free, then the natural boundary condition
∂3L [y∗; ζ∗]β(b) =− ∫ b a ∂5L [y∗; ζ∗]β(t) dβt holds. Proof
Letη∈ Yandδ∈R be arbitrary. A necessary condition for(y∗, ζ∗)to be an extremizer is given by
ϕ′(0) = 0
whereϕ(ϵ) =L[y∗+ ϵη, ζ∗+ ϵδ]. By Lemma4, we conclude that ∫ b a ( ∂2L [y∗; ζ∗]β(t)· η (β (t)) + ∂3L [y∗; ζ∗]β(t)· Dβ[η] (t) + ∂4L [y∗; ζ∗]β(t)· η(a) + ∂5L [y∗; ζ∗]β(t)· η(b) + ∂6L [y∗; ζ∗]β(t)· δ ) dβt = 0 and using integration by parts we obtain
∂3L [y∗; ζ∗]β(t)· η (t) b a + ∫ b a ( ∂2L [y∗; ζ∗]β(t)− Dβ[∂3L] [y∗; ζ∗]β(t) ) dβt + ∫ b a ( ∂4L [y∗; ζ∗]β(t)· η(a) + ∂5L [y∗; ζ∗]β(t)· η(b) ) dβt + ∫ b a ∂6L [y∗; ζ∗]β(t)· δ dβt = 0.
Since no boundary conditions are imposed,η do not need to vanish at the endpoints. However, this last equation must be satisfied for allη∈ Yand in particular for the functions that do vanish at the endpoints. Using the same arguments used in the proof of Theorem5we conclude that(y∗, ζ∗)satisfies the general Euler-Lagrange equation
∂2L [y∗; ζ∗]β(t) = Dβ[∂3L] [y∗; ζ∗]β(t) , t∈ [a, b]β
and condition ∫ b
a
∂6L [y∗; ζ∗]β(t) dβt = 0.
1. Suppose thaty(a)is free. Ify(b)is fixed, thenη(b) = 0; ify(b)is free then we can restrict to all functions
η∈ Ysuch thatη(b) = 0. In this case, using general quantum Euler-Lagrange equation and takingδ = 0, we conclude that
∂3L [y∗; ζ∗]β(a)· η (a) = ∫ b
a
∂4L [y∗; ζ∗]β(t)· η(a) dβt. From the arbitrariness ofηit follows that
∂3L [y∗; ζ∗]β(a) = ∫ b
a
2. Suppose thaty(b)is free. Ify(a)is fixed, thenη(a) = 0; ify(a)is free then we can restrict to all functions
η∈ Ysuch thatη(a) = 0. In this case, using general quantum Euler-Lagrange equation and choosingδ = 0, it follows that∂3L [y∗; ζ∗]β(b) =−
∫ b a
∂5L [y∗; ζ∗]β(t) dβt. The proof is complete.
For the special case whereβ(t) = qt + ω, for someq∈]0, 1[andω>0, and the LagrangianLdoes not depend on the real parameterζ, we obtain the following result of the Hahn variational calculus, where we suppose thatL
satisfies conditions (H1)-(H3) and (i)-(iii) of Lemma4adapted for the case thatLdepends only on the arguments
(t, y (qt + ω) , Dq,ω[y](t), y(a), y(b)).In order to simplify the expressions, we denote
[y]q,ω(t) := (t, y (qt + ω) , Dq,ω[y](t), y(a), y(b)) and
[a, b]q,ω:={qna + ω[n]q} ∪ {qnb + ω[n]q} ∪ {ω0} (see Remark1for more details).
Corollary 2 ([19])
Ify∗is a local extremizer for problem (2), theny∗satisfies the quantum Euler-Lagrange equation
∂2L [y]q,ω(t) = Dq,ω[∂3L] [y]q,ω(t), t∈ [a, b]q,ω. Moreover,
1. ify (a)is free, then the natural boundary condition
∂3L [y]q,ω(a) = ∫ b
a
∂4L [y]q,ω(t) dq,ωt holds;
2. ify (b)is free, then the natural boundary condition
∂3L [y]q,ω(b) =− ∫ b
a
∂5L [y]q,ω(t) dq,ωt holds.
In the particular case where the Lagrangian functionLdoes not depend on the state valuesy(a)andy(b), and on the real parameterζ, and under appropriate assumptions on the Lagrangian, we obtain the following necessary optimality conditions.
Corollary 3
Ify∗is a local extremizer to problem
L [y] =
∫ b a
L (t, y (β (t)) , Dβ[y] (t)) dβt−→extremize, theny∗satisfies the Euler-Lagrange equation
Dβ[∂3L] (t, y (β (t)) , Dβ[y] (t)) = ∂2L (t, y (β (t)) , Dβ[y] (t)) for allt∈ [a, b]βand
1. ify (a)is free, then the natural boundary condition
∂3L (t, y (β (t)) , Dβ[y] (t)) (a) = 0 holds;
2. ify (b)is free, then the natural boundary condition
∂3L (t, y (β (t)) , Dβ[y] (t)) (b) = 0 holds.
Remark 5
Note that the sufficient condition for optimality presented in Theorem14is still valid for variational problems with free boundary conditions.
3.2. Generalized quantum isoperimetric problems
In this section we consider problem (2) when we are in presence of an integral constraint. This kind of variational problems is know in the literature as isoperimetric problems. The problem that consist in the search of a closed plane curve of a given perimeter which encloses the greatest area is a well-known example of an isoperimetric problem.
The quantum isoperimetric problem that we consider here consists in extremizing the functional
L [y, ζ] =
∫ b a
L (t, y (β (t)) , Dβ[y] (t) , y(a), y(b), ζ) dβt (8) in the class of functionsy∈ Ysatisfying the boundary conditions (3) and the integral constraint
I [y, ζ] =
∫ b a
F (t, y (β (t)) , Dβ[y] (t) , y(a), y(b), ζ) dβt = τ (9) for some fixed valueτ ∈R.
Before presenting necessary optimality conditions for such kind of variational problems, we first give some basic definitions.
Definition 16
We say that a pair (y∗, ζ∗)is a local minimizer (resp. local maximizer) to the isoperimetric problem (8)-(9) if there existsγ > 0such thatL [y∗, ζ∗]6L [y, ζ](resp.L [y∗, ζ∗]>L [y, ζ]) for all functionsy∈ Y and allζ∈R
satisfying the integral constraint (9) with∥y∗− y∥ < γand|ζ∗− ζ| < γ.
Definition 17
We say that(y, ζ) is a normal extremizer to the isoperimetric problem (8)-(9) if (y, ζ)is a local extremizer to problem (8)-(9) that is not an extremal to functionalI; if(y, ζ)is a local extremizer and an extremal to functional
Iwe say that(y, ζ)is an abnormal extremizer.
Now we are ready to prove necessary optimality conditions for isoperimetric problems, with and without fixed boundary conditions, for the particular case of normal extremizers.
Theorem 18 (Necessary optimality conditions for normal extremizers to isoperimetric problems)
Let us suppose thatLandFsatisfy hypotheses (H1)-(H3) and (i)-(iii) of Lemma4. If(y∗, ζ∗)is a normal extremizer to the isoperimetric problem (8)-(9), then there exists a real numberλ∈R such that(y∗, ζ∗)satisfies the following conditions
∂2H [y; ζ]β(t) = Dβ[∂3H] [y; ζ]β(t) , t∈ [a, b]β
and ∫
b
a
∂6H [y; ζ]β(t) dβt = 0, whereH = L− λF.Moreover,
1. ify(a)is free, then
∂3H [y∗; ζ∗]β(a) = ∫ b
a
∂4H [y∗; ζ∗]β(t) dβt; 2. ify(b)is free, then
∂3H [y∗; ζ∗]β(b) =− ∫ b
a
∂5H [y∗; ζ∗]β(t) dβt.
Proof
Let us suppose that(y∗, ζ∗)is a normal extremizer for the isoperimetric problem (8)-(9). Let us define two real functionsϕ, ψ :R2→R by
ϕ (ε1, ε2) = L [y∗+ ε1η1+ ε2η2, ζ∗+ ε1δ1+ ε2δ2]
ψ (ε1, ε2) = I [y∗+ ε1η1+ ε2η2, ζ∗+ ε1δ1+ ε2δ2]− τ,
whereη1is an arbitrary variation,η2is a fixed variation (that we will choose later),δ1is an arbitrary real number andδ2is a fixed real number (that we will choose later).
Note that ∂2ψ (0, 0) = ∫ b a ( ∂2F [y∗; ζ∗] (t)· η2(β (t)) + ∂3F [y∗; ζ∗] (t)· Dβ[η2] (t) +∂4F [y∗, ζ∗] (t)· η2(a) + ∂5F [y∗; ζ∗] (t)· η2(b) +∂6F [y∗; ζ∗] (t)· δ2 ) dβt. Using integration by parts formula we get that
∂2ψ (0, 0) = ∫ b a ( ∂2F [y∗; ζ∗] (t)− Dβ[∂3F ] [y∗; ζ∗]β(t) ) · η2(β (t)) dβt + ∫ b a ( ∂4F [y∗; ζ∗] (t)· η2(a) + ∂5F [y∗; ζ∗] (t)· η2(b) + ∂6F [y∗; ζ∗] (t)· δ2 ) dβt + ∂3F [y∗; ζ∗] (t)· η2(b)− ∂3F [y∗; ζ∗] (t)· η2(a). (10) Since(y∗, ζ∗)is not an extremal ofI, then it can happen three cases:
(i) ∂2F [y∗; ζ∗]β(t)̸= Dβ[∂3F ] [y∗; ζ∗]β(t)and ∫ b a ∂6F [y∗; ζ∗]β(t) dβt = 0; (ii) ∂2F [y∗; ζ∗]β(t) = Dβ[∂3F ] [y∗; ζ∗]β(t)and ∫ b a ∂6F [y∗; ζ∗]β(t) dβt̸= 0; (iii) ∂2F [y∗; ζ∗]β(t)̸= Dβ[∂3F ] [y∗; ζ∗]β(t)and ∫ b a ∂6F [y∗; ζ∗]β(t) dβt̸= 0.
Let us suppose we are in case (i). If we restrictη2to those functions such thatη2(a) = η2(b) = 0, we get from (10) that ∂2ψ (0, 0) = ∫ b a ( ∂2F [y∗; ζ∗]β(t)− Dβ[∂3F ] [y∗; ζ∗]β(t) ) · η2(β (t)) dβt.
Since ∂2F [y∗; ζ∗]β(t)− Dβ[∂3F ] [y∗; ζ∗]β(t)̸= 0, then we can choose η2 such that ∂2ψ (0, 0)̸= 0. Since
ψ (0, 0) = 0, then by the Implicit Function Theorem there exists a function h, defined in a neighborhoodV of
0such thath (0) = 0andψ (ε1, h (ε1)) = 0for anyε1∈ V. This means that there exists a family of pairs(y, ζ)
where {
y = y∗+ ε1η1+ h(ε1)η2
that satisfies the isoperimetric constraint. Moreover,(0, 0)is an extremizer ofϕsubject to the constraintψ = 0and
∇ψ (0, 0) ̸= (0, 0) .Therefore, there is a Lagrange multiplierλ∈R such that ∇ϕ (0, 0) = λ∇ψ (0, 0) .
Restrictingη1to those such thatη1(a) = η1(b) = 0we conclude that
∂1ϕ (0, 0) = ∫ b a ( ∂2L [y∗; ζ∗]β(t)− Dβ[∂3L] [y∗; ζ∗]β(t) ) · η1(β (t)) dβt and ∂1ψ (0, 0) = ∫ b a ( ∂2F [y∗; ζ∗]β(t)− Dβ[∂3F ] [y∗; ζ∗]β(t) ) · η1(β (t)) dβt. Therefore ∫ b a ( (∂2L [y∗; ζ∗]β(t)− Dβ[∂3L] [y∗; ζ∗]β(t))− λ(∂2F [y∗; ζ∗]∗(t)− Dβ[∂3F ] [y∗; ζ∗]β(t)) ) · η1(β (t)) dβt = 0, and Lemma2proves that
∂2L [y∗; ζ∗]∗(t)− Dβ[∂3L] [y∗; ζ∗]β(t) = λ(∂2F [y∗; ζ∗] (t)− Dβ[∂3F ] [y∗; ζ∗]β(t)) for allt∈ [a, b]β. Then
∂2H [y∗; ζ∗]β(t) = Dβ[∂3H] [y∗; ζ∗]β(t)
forH = L− λF and allt∈ [a, b]β, proving thatH satisfies the general quantum Euler-Lagrange equation. Since, by hypothesis, ∫ b a ∂6L [y∗; ζ∗]β(t) dβt = 0 and ∫ b a ∂6F [y∗; ζ∗]β(t) dβt = 0
then it is obvious that ∫
b
a
∂6H [y∗; ζ∗]β(t) dβt = 0.
Let us now consider the case (ii). If in equality (10) we restrictη2to those such thatη2(a) = η2(b) = 0we get
∂2ψ(0, 0) = ∫ b a ∂6F [y∗; ζ∗]β· δ2dβt. Since ∫ b a ∂6F [y∗; ζ∗]β(t) dβt̸= 0
we can chooseδ2such that∂2ψ(0, 0)̸= 0. Applying the Implicit Function Theorem and the Lagrange multiplier rule, is possible to prove that there exists a real constantλsuch that forH = L− λF, we have
∫ b a ∂6H[y∗; ζ∗]βdβt = 0. Since, ∂2F [y∗; ζ∗]β(t)− Dβ[∂3F ] [y∗; ζ∗]β(t) = 0, t∈ [a, b]β and ∂2L [y∗; ζ∗]β(t)− Dβ[∂3L] [y∗; ζ∗]β(t) = 0, t∈ [a, b]β
it follows that
∂2H [y∗; ζ∗]β(t)− Dβ[∂3H] [y∗; ζ∗]β(t) = 0, t∈ [a, b]β.
Finally, suppose we are in case (iii). The proof that(y∗, ζ∗)satisfies the Euler-Lagrange equation with respect toH is similar to the case (i), if we consider in this caseδ2= 0. The proof that (y∗, ζ∗)satisfies the equality ∫ b
a
∂6H [y; ζ]β(t) dβt = 0is similar to the case (ii), considering hereη2= 0. Now we will prove the natural boundary conditions.
1. Suppose thaty(a)is free. Ify(b)is fixed, thenη1(b) = 0; ify(b)is free then we can restrict to all functions
η1∈ Y such thatη1(b) = 0. Therefore,
∂1ϕ (0, 0) = ∫ b a ( ∂2L [y∗; ζ∗]β(t)− Dβ[∂3L] [y∗; ζ∗]β(t) ) · η1(β (t)) dβt + ∫ b a ( ∂4L [y∗; ζ∗]β(t)· η1(a) + ∂6L [y∗; ζ∗]β· δ2 ) dβt − ∂3L [y∗; ζ∗]β(a)· η1(a) and ∂1ψ (0, 0) = ∫ b a ( ∂2F [y∗; ζ∗]β(t)− Dβ[∂3F ] [y∗; ζ∗]β(t) ) · η1(β (t)) dβt + ∫ b a ( ∂4F [y∗; ζ∗]β(t)· η1(a) + ∂6F [y∗; ζ∗]β· δ2 ) dβt − ∂3F [y∗; ζ∗]β(a)· η1(a) .
Since ∇ϕ (0, 0) = λ∇ψ (0, 0) , H satisfies the general quantum Euler-Lagrange equation and ∫ b a ∂6H [y∗; ζ∗]β(t) dβt = 0,then ∂3H [y∗; ζ∗]β(a)· η1(a) = ∫ b a ∂4H [y∗; ζ∗]β(t)· η1(a) dβt. From the arbitrariness ofη1it follows that
∂3H [y∗; ζ∗]β(a) = ∫ b
a
∂4H [y∗; ζ∗]β(t) dβt.
2. Suppose now thaty(b)is free. Using similar arguments as the ones used in 1. we prove that
∂3H [y∗; ζ∗]β(b) =− ∫ b
a
∂5H [y∗; ζ∗]β(t) dβt. The proof is complete.
The previous result can be generalized for the case of abnormal extremizers to isoperimetric problems as follows.
Theorem 19 (Necessary optimality conditions for normal and abnormal extremizers to isoperimetric problems)
Let us suppose thatLandF satisfy hypotheses (H1)-(H3) and (i)-(iii) of Lemma4. If(y∗, ζ∗)is an extremizer to the isoperimetric problem (8)-(9), then there exist two constantsλ0, λ∈R, not both zero, such that(y∗, ζ∗)satifies the conditions
and ∫ b
a
∂6H [y; ζ]β(t) dβt = 0, whereH = λ0L− λF.Moreover,
1. ify(a)is free, then
∂3H [y∗; ζ∗]β(a) = ∫ b
a
∂4H [y∗; ζ∗]β(t) dβt; 2. ify(b)is free, then
∂3H [y∗; ζ∗]β(b) =− ∫ b
a
∂5H [y∗; ζ∗]β(t) dβt.
Proof
The proof is similar to the proof of Theorem18, but in this case we apply the abnormal Lagrange multiplier rule. This result ensures the existence of two constantsλ0 andλ, not both zero, such thatλ0∇ϕ (0, 0) = λ∇ψ (0, 0) . The rest of the proof is straightforward, so it will be omitted.
In the case whereL and F do not depend on y(a), y(b)and ζ, and under appropriate assumptions on those functionsLandF, we obtain the following result for the simplest quantum isoperimetric problem.
Corollary 4
Ify∗is a local extremizer to problem
L [y] =
∫ b a
L (t, y (β (t)) , Dβ[y] (t)) dβt−→extremize, subject to the integral constraint
I [y] =
∫ b a
F (t, y (β (t)) , Dβ[y] (t)) dβt = τ, (11) then there exist two constantsλ0, λ∈R, not both zero, such thaty∗satisfies the quantum Euler-Lagrange equation
∂2H (t, y(β(t)), Dβ[y] (t)) = Dβ[∂3H] (t, y(β(t)), Dβ[y] (t)) , for allt∈ [a, b]β, whereH = λ0L− λF.Moreover,
1. ify(a)is free, then∂3H (a, y(β(a)), Dβ[y] (a)) = 0; 2. ify(b)is free, then∂3H (b, y(β(b)), Dβ[y] (b)) = 0.
Remark 6
Again, if we restrict ourselves to the particular case whereβ(t) = qt + ω, for someq∈]0, 1[andω>0, Theorem 29 and Theorem 30 generalize, respectively, Theorem 3.9 and Theorem 3.10 of [19] for the Hahn calculus of variations.
4. Illustrative examples
In this section we present some examples in order to illustrate our results. Although, to the best of our knowledge, a general result guarantying the existence of solutions of nonlinear second orderβ-difference equations, as well as a method to construct these solutions, have not yet been presented, we show here one example where it is possible to exhibit one solution to this kind of difference equations. For the particular case of second order linear homogeneous
Example 1
Considerβ(t) = ln t + 1, whereI = [1, +∞[. We remark thatβ is a strictly increasing and continuous function that has a unique fixed point that iss0= 1. Moreover,β satisfies inequality (1) where the equality holds only if
t = s0. Note also thatβ(t)6tfor allt∈ I. Consider the following problem
L[y] =
∫ 3 1
(Dβ[y])2− 1)2· (y(β(t)))2dβt−→ minimize over ally∈ Ysatisfying the boundary conditionsy(1) = 0andy(3) = 3.
The quantum Euler-Lagrange equation (7) takes the form
2Dβ [( (Dβ[y](t))2− 1 ) · (y(β(t)))2· D β[y](t) ] (t) = y(β(t))· ( (Dβ[y](t))2− 1 )2 . (12) Consider y∗(t) = 0 if t∈ [1, 2] t if t∈ ]2, +∞[ . Note that Dβ[y∗](t) = 0 if t∈ [1, 2] t t−β(t) if t > 2 ∧ β(t)62 1 if t > 2∧ β(t) > 2
and thereforey∗ is an admissible function satisfying (12) (we remark that for alltsuch thatt > 2andβ(t)62,
y(β(t)) = 0).
Noting thats0= 1, we can conclude by Theorem5thatL[y]>0. SinceL[y∗] = 0, it follows thaty∗is indeed a minimizer. Note that the minimizery∗is not continuous while in the classical calculus of variations we deal with functions that are necessarily continuous.
Example 2
Letβ :R→R be such thatβ (t) = qt + ωforq∈ ]0, 1[andω>0. We remark thatβhas a unique fixed point that iss0=
ω
1− q. Consider the following problem L [y, ζ] =
∫ 1 0
L[y; t]βdβt−→ minimize where
L[y; t]β= 2y(qt + ω) + (Dβ[y] (t)) 2
+ (y(1)− 2)2+ (1− ζ)2
in the class of admissible functions y∈ Y that satisfy the boundary condition y(0) = 0. If (y∗, ζ∗) is a local minimizer to this problem, then, by Theorem15,(y∗, ζ∗)satisfies the following conditions:
(i) Dβ[∂3L] [y∗; ζ∗]β(t) = ∂2L [y∗; ζ∗]β(t)⇔ Dβ[Dβ[y∗]] (t) = 1; (ii) ∫ 1 0 ∂6L [y∗; ζ∗]β(t) dβt = 0⇔ ζ∗= 1; (iii) ∂3L [y∗; ζ∗]β(1) =− ∫ 1 0 ∂5L [y∗; ζ∗]β(t) dβt⇔ Dβ[y∗](1) = 2− y∗(1). It is easy to check that
y∗(t) = 1 q + 1t 2− ( ω q + 1− c ) t + d
wherec, d∈R, is a solution to the Euler-Lagrange equation given in(i). Sincey∗(0) = 0, we conclude thatd = 0. Observing that Dβ[y∗](t) = 1 q + 1(t + qt + ω)− ( ω q + 1− c ) ,
then the natural boundary condition(iii)gives
c = q + ω 2(q + 1).
Hence, the pair(y∗, 1)where
y∗(t) = 1 q + 1t 2− ( ω q + 1 − q + ω 2(q + 1) ) t
is a candidate to be a local minimizer for the problem. Moreover, since the Lagrangian function is jointly convex, Theorem14guarantees that(y∗, 1)is a global minimizer.
Example 3
Suppose now that we want to minimize the following functional
L [y, ζ] =
∫ 1 0
(
2y(qt + ω) + (Dβ[y] (t))2+ ζy2(0) + (y(1)− 2)2+ (1− ζ)2 )
dβt whereβ (t) = qt + ωforq∈ ]0, 1[andω>0, in the class of admissible functionsy∈ Yandζ∈R.
If(y∗, ζ∗)is a local minimizer to this problem, then, by Theorem15, we conclude that:
(i) Dβ[∂3L] [y∗; ζ∗]β(t) = ∂2L [y∗; ζ∗]β(t)⇔ Dβ[Dβ[y∗]] (t) = 1; (ii) ∫ 1 0 ∂6L [y∗; ζ∗]β(t) dβt = 0; (iii) ∂3L [y∗; ζ∗]β(0) = ∫ 1 0 ∂4L [y∗; ζ∗]β(t) dβt; (iv) ∂3L [y∗; ζ∗]β(1) =− ∫ 1 0 ∂5L [y∗; ζ∗]β(t) dβt. Again, it is easy to check that
y∗(t) = 1 q + 1t 2− ( ω q + 1− c ) t + d
wherec, d∈R, is a solution to the Euler-Lagrange equation given in(i). In order to determine the values ofc, d, ζ∗
we will use condition(ii)and the natural boundary conditions(iii)− (iv). Using condition(ii)we conclude that
ζ∗=d
2+ 2
2 .
The natural boundary condition(iii)allows us to conclude thatc = d
3+ 2d
2 . Finally, using the natural boundary
condition(iv)we get thatd(d2+ 3) = q + ω
q + 1. Hence, the pair
( y∗,d 2+ 2 2 ) , where y∗(t) = 1 q + 1t 2− ( ω q + 1− d3+ 2d 2 ) t + d and d(d2+ 3) = q + ω q + 1,
is a candidate to be a local minimizer for the problem. Note that if we restrict our problem for the case where the parameterζbelongs toR+, then the Lagrangian functionLis jointly convex, and therefore, from Theorem14, we can conclude that
( y∗,d 2+ 2 2 ) is a global minimizer.
5. Conclusion and perspectives
In this paper we introduced the quantum variational calculus based on the general quantum differential operatorDβ. We proved necessary optimality conditions of Euler-Lagrange type and natural boundary conditions for generalized problems of the calculus of variations where the Lagrangian function not only depend on timet∈ [a, b], function
y and the quantum derivativeDβ[y], but also on a real parameterζand the state valuesy(a) andy(b). We also provided a sufficient optimality condition.
There is still a lot of work to do in order to develop this new variational calculus. For example, one interesting direction is to extend our results to the order case. In our opinion, the main difficulty is to prove the higher-order fundamental lemma of the calculus of variations because of the generality of the functionβ(see Lemma 3.8 of [8] for the particular case of Hahn’s variational calculus). Another open question is to consider problems of the general quantum variational calculus with time delay. Here, the main difficulty is that the chain rule, as known from classical calculus, does not hold for the general quantum derivative.
Acknowledgments
This work was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (FCT), within project UID/MAT/04106/2013.
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