Euler-Lagrange equations for composition functionals in calculus of variations on time scales

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H(\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.


1.
Introduction. The calculus on time scales was introduced by Bernd Aulbach and Stefan Hilger in 1988 [6]. The new theory bridges the divide and extends the traditional areas of continuous and discrete analysis and the various dialects of qcalculus [14] into a single theory [11,12,20]. The calculus of variations on time scales was born with the works [2,8,18] and has interesting applications in Economics [3,4,5,15,26]. Currently, several researchers are getting interested in the new theory and contributing to its development (see, e.g., [7,9,10,16,21,22,23,24,25]). The present work is dedicated to the study of general (non-classical) problems of calculus of variations on an arbitrary time scale T. As a particular case, by choosing T = R, one gets the generalized calculus of variations [13] with functionals of the form where f has n components and H has n independent variables. Cases of calculus of variations as these appear in practical applications (see [13] and the references given therein) but cannot be solved using the classical theory. Therefore, an extension of this theory is needed.
The paper is organized as follows. In Section 2, some preliminaries on time scales are presented. Our results are given in Section 3 and Section 4. We begin Section 3 by formulating the general (non-classical) problem of calculus of variations (1) on an arbitrary time scale. We obtain a general formula for the Euler-Lagrange equations and natural boundary conditions for the general problem (Theorem 3.2), which are then applied to the product (Corollary 3.4) and the quotient (Corollary 3.7). In Section 4 we prove a necessary optimality condition for the general isoperimetric problem (Theorem 4.3 and Theorem 4.5). Throughout the paper several examples illustrating the new results are discussed in detail.
2. Preliminaries. The following definitions and theorems will serve as a short introduction to the calculus of time scales; they can be found in [11,12].
A nonempty closed subset of R is called a time scale and it is denoted by T. The real numbers (R), the integers (Z), the natural numbers (N), the h-numbers (hZ := {hz|z ∈ Z}, where h > 0 is a fixed real number), and the q-numbers (q N0 := {q k |k ∈ N 0 }, where q > 1 is a fixed real number) are examples of time scales, as are {0, 1 2 , 1}, [2,3] ∪ N, and [−1, 1] ∪ [2,3], where [−1, 1] and [2,3] are real number intervals. We assume that a time scale T has the topology that it inherits from the real numbers with the standard topology.
while the backward jump operator ρ : T → T is defined by ρ(t) = sup {s ∈ T : s < t}, for all t ∈ T.
In this definition we consider σ(M ) = M if T has a maximum M and ρ(m) = m if T has a minimum m.
A point t ∈ T is called right-dense, right-scattered, left-dense and left-scattered if σ(t) = t, σ(t) > t, ρ(t) = t and ρ(t) < t, respectively. Points that are simultaneously right-scattered and left-scattered are called isolated. Points that are simultaneously right-dense and left-dense are called dense.

Definition 2.3.
A time scale T is called regular if the following two conditions are satisfied: (i) σ(ρ(t)) = t, for all t ∈ T; and (ii) ρ(σ((t)) = t, for all t ∈ T.
Following [11], let us define Definition 2.4. We say that a function f : T → R is delta differentiable at t ∈ T κ if there exists a number f ∆ (t) such that for all ε > 0 there is a neighborhood U of t (i.e., U = (t − δ, t + δ) ∩ T for some δ > 0) such that We call f ∆ (t) the delta derivative of f at t and f is said delta differentiable on T κ provided f ∆ (t) exists for all t ∈ T κ .
is not uniquely defined, since for such a point t, small neighborhoods U of t consist only of t and, besides, we have σ(t) = t. For this reason, maximal left-scattered points are omitted in Definition 2.4.
Note that in right-dense points f ∆ (t) = lim s→t , provided this limit exists, and in right-scattered points f , provided f is continuous at t.
i.e., the delta derivative coincides with the usual one.
i.e., we get the usual derivative of quantum calculus [19].
A function f : T → R is called rd-continuous if it is continuous at right-dense points and if its left-sided limit exists at left-dense points. We denote the set of all rd-continuous functions by C rd and the set of all delta differentiable functions with rd-continuous derivative by C 1 rd . Now we introduce the concept of integral for a function f : Open intervals and half-open intervals in T are defined accordingly. In what follows all intervals will be time scale intervals.
It is known that rd-continuous function possess an antiderivative, i.e., there exists a function F with F ∆ = f , and in this case the delta integral is defined by for all a, b ∈ T.
The delta integral has the following properties: (i) if f ∈ C rd and t ∈ T κ , then The Dubois-Reymond lemma of the calculus of variations on time scales will be useful for our purposes.
rd . Using parentheses around the end-point conditions means that these conditions may or may not be present. We assume that: (i) the function H : R n → R has continuous partial derivatives with respect to its arguments and we denote them by H ′ i , i = 1, . . . , n; . . , n, have partial continuous derivatives with respect to y, v for all t ∈ [a, b] and we denote them by f iy , f iv ; rd is said to be an admissible function provided that it satisfies the end-points conditions (if any is given).
Let us consider the following norm in C 1 rd : Definition 3.1. An admissible functionx is said to be a weak local minimizer (respectively weak local maximizer ) for (1) Next theorem gives necessary optimality conditions for problem (1).
Ifx is a weak local solution of the problem (1), then the Euler-Lagrange equation and if x(b) is not specified, then Proof. Suppose that L[x] has a weak local extremum atx. For an admissible vari- , respectively, is free (it is possible that both are free). A necessary condition forx to be an extremizer for L[x] is given by φ ′ (ε)| ε=0 = 0. Using the chain rule for obtaining the derivative of a composed function we get The necessary condition φ ′ (ε)| ε=0 = 0 can be written as In particular, equation (5) for some c ∈ R and all t ∈ [a, b]. Hence, equation (2) holds for all t ∈ [a, b] κ . Equation (5) must be satisfied for all admissible values of h(a) and h(b). Consequently, equations (5) and (6) imply that From the properties of the delta integral and from (6), it follows that If x(t) is not preassigned at either end-point, then h(a) and h(b) are both completely arbitrary and we conclude that their coefficients in (7) must each vanish. It follows that condition (3) holds when x(a) is not given, and condition (4) holds when x(b) is not given.
Therefore (4) can be written in the form Choosing T = R in Theorem 3.2 we immediately obtain Theorem 3.1 and Equation (4.1) in [13]. The Euler-Lagrange Equation for the product functional can be deduced from Theorem 3.2.
Ifx is a local minimum of (8), then the Euler-Lagrange equation must hold, i.e., where If Q 2 = 0, then also Q 1 = 0. This contradicts the fact that on any time scale a global minimizer for the problem Hence, Q 2 = 0 and equation (9) implies that candidate solutions for problem (8) are those satisfying the delta differential equatioñ subject to boundary conditions x(0) = 0 and x(1) = 1. Solving equation (10) we obtain Therefore, a solution of (10) depends on the time scale. Let us consider, for example, T = R and T = 0, 1 2 , 1 . On T = R we obtain Substituting (11) into functionals F 1 and F 2 gives Solving the system of equations (12) we obtain Therefore,x (t) = −t 2 + 2t is a candidate extremizer for problem (8) on T = R. Note that nothing can be concluded as to whetherx gives a minimum, a maximum, or neither of these for L. The solution of (10) on T = 0, 1 2 , 1 is Constants Q 1 and Q 2 are determined by substituting (13) into functionals F 1 and F 2 . The resulting system of equations is Since system of equations (14) has no real solutions, we conclude that there exists no extremizer for problem (8) on T = 0, 1 2 , 1 among the set of functions that we consider to be admissible.
Assuming that the denominator does not vanish, the Euler-Lagrange equation for the quotient problem can be deduced from Theorem 3.2.
The Euler-Lagrange equation for this problem is where Q is the value of functional L in a solution of (16). Since Q = 0, it follows that Solving equation (17) we obtain Therefore, a solution of (17) depends on the time scale. Let us consider, for example, T = R and T = {0, 1 2 , 1}. On T = R we obtain Substituting (18) into functional L yields Solving equation (19) we obtain Q ∈ 1 4 − √ 3 . Therefore, is a candidate local minimizer while is a candidate local maximizer for problem (16) on T = R. The solution of (17) on T = 0, 1 2 , 1 is The constant Q is determined by substituting (20) into L. The resulting equation is 1 4 Solving (21)  and stationary functions are and  Therefore (22) is a candidate local minimizer while (23) is a candidate local maximizer for problem (16) on T = 0, 1 2 , 1 . Example 3.11. Consider the problem where q : [a, b] → R is a continuous function. The Euler-Lagrange equation for this problem is where Q is the value of functional L in a solution of (24). It is easily seen that (25)-(26) is a case of the Sturm-Liouville eigenvalue problem on time scales (see [1] and [17]). It follows that the problem of determining eigenfunctions of (25) subject to (26) is equivalent to the problem of determining functions satisfying (26) which render L stationary.
4. Isoperimetric problems. Let us consider now the general (non-classical) isoperimetric problem on time scales. The problem consists of minimizing or maximizing in the class of functions x ∈ C 1 rd satisfying the boundary conditions and the constraint where x a , x b , k are given real numbers. We assume that: (i) functions H : R n → R and P : R m → R have continuous partial derivatives with respect to their arguments and we denote them by H ′ i , i = 1, . . . , n, and P ′ i , i = 1, . . . , m; (ii) functions (t, y, v) → f i (t, y, v), i = 1, . . . , n, and (t, y, v) → g j (t, y, v), j = 1, . . . , m, from [a, b] × R 2 to R have partial continuous derivatives with respect to y, v for all t ∈ [a, b] and we denote them by f iy , f iv and g jy , g jv ; (iii) f i , f iy , f iv , i = 1, . . . , n, and g j , g jy , g jv , j = 1, . . . , m, are rd-continuous in t for all x ∈ C 1 rd . Definition 4.1. An admissible functionx is said to be a weak local minimizer (respectively weak local maximizer ) for the isoperimetric problem (27) where (•) = t,x σ (t),x ∆ (t) and (•) = τ,x σ (τ ),x ∆ (τ ) , for some constant c and for all t ∈ [a, b]. An extremizer (i.e., a weak local minimizer or a weak local maximizer) for the problem (27)-(29) that is not an extremal for K is said to be a normal extremizer; otherwise (i.e., if it is an extremal for K), the extremizer is said to be abnormal.
Proof. Consider a variation ofx, sayx =x + ε 1 h 1 + ε 2 h 2 , where h i ∈ C 1 rd and h i (a) = h i (b) = 0, i = 1, 2, and ε i is a sufficiently small parameter (ε 1 and ε 2 must be such that x −x 1 < δ for some δ > 0). Here, h 1 is an arbitrary fixed function and h 2 is a fixed function that will be chosen later. Define the real function We have = 0. SinceK(0, 0) = 0, by the implicit function theorem we conclude that there exists a function ε 2 defined in the neighborhood of zero, such thatK(ε 1 , ε 2 (ε 1 )) = 0, i.e., we may choose a subset of variationsx satisfying the isoperimetric constraint.