Strong Minimizers of the Calculus of Variations on Time Scales and the Weierstrass Condition

We introduce the notion of strong local minimizer for the problems of the calculus of variations on time scales. Simple examples show that on a time scale a weak minimum is not necessarily a strong minimum. A time scale form of the Weierstrass necessary optimality condition is proved, which enables to include and generalize in the same result both continuous-time and discrete-time conditions.


Introduction
Dynamic equations on time scales is a recent subject that allows the unification and extension of the study of differential and difference equations in one and same theory [10].
The calculus of variations on time scales was introduced in 2004 with the papers of Martin Bohner [6] and Roman Hilscher and Vera Zeidan [15]. Roughly speaking, in [6] the basic problem of the calculus of variations on time scales with given boundary conditions is introduced, and time scale versions of the classical necessary optimality conditions of Euler-Lagrange and Legendre proved, while in [15] necessary conditions as well as sufficient conditions for variable end-points calculus of variations problems on time scales are established. Since the two pioneer works [6,15] and the understanding that much remains to be done in the area [13], several recent studies have been dedicated to the calculus of variations on time scales: the time scale Euler-Lagrange equation was proved for problems with double delta-integrals [9] and for problems with higher-order delta-derivatives [14]; a correspondence between the existence of variational symmetries and the existence of conserved quantities along the respective Euler-Lagrange delta-extremals was established in [5]; optimality conditions for isoperimetric problems on time scales with multiple constraints and Pareto optimality conditions for multiobjective delta variational problems were studied in [20]; a weak maximum principle for optimal control problems on time scales has been obtained in [16]. Such results may also be formulated via the nabla-calculus on time scales, and seem to have interesting applications in economics [1,2,3,21].
In all the works available in the literature on time scales the variational extrema are regarded in a weak local sense. Differently, here we consider strong solutions of problems of the calculus of variations on time scales. In Section 2 we briefly review the necessary results of the calculus on time scales. The reader interested in the theory of time scales is referred to [10,11], while for the classical continuous-time calculus of variations we refer to [12,19], and to [18] for the discrete-time setting. In Section 3 the concept of strong local minimum is introduced (cf. Definition 3.1), and an example of a problem of the calculus of variations on the time scale T = { 1 n : n ∈ N} ∪ {0} is considered showing that the standard weak minimum used in the literature on time scales is not necessarily a strong minimum (cf. Example 3.2). Our main result is a time scale version of the Weierstrass necessary optimality condition for strong local minimum (cf. Theorem 3.3). We end with Section 4, illustrating our main result with the particular cases of discrete-time and the q-calculus of variations [4].

Time Scales Calculus
In this section we introduce basic definitions and results that will be needed for the rest of the paper. For a more general theory of calculus on time scales, we refer the reader to [10,11].
A nonempty closed subset of R is called a time scale and it is denoted by T. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t we say that t is left-scattered. Also, if t < sup T and σ(t) = t, than t is called right-dense, and if t > inf T and ρ(t) = t, then t is called left-dense. The set T κ is defined as T without the left-scattered maximum of T (in case it exists).
The graininess function µ : if it is regulated and if it is continuous at all right-dense points t ∈ T. Following [15], a function f is piecewise rd-continuous (we write f ∈ C prd ) if it is regulated and if it is rd-continuous at all, except possibly at finitely many, right-dense points t ∈ T.
We say that a function f : We call f △ (t) the delta derivative of f at t and say that f is delta differentiable , provided this limit exists, and in right-scattered i.e., we get the usual derivative of Quantum calculus [17].
Let f, g : T → R be delta differentiable at t ∈ T κ . Then (see, e.g., [10]), where we abbreviate here and throughout the text f • σ by f σ .
prd ) if f is continuous and f △ exists for all, except possibly at finitely many, t ∈ T κ and f △ ∈ C rd . It is known that piecewise rd-continuous functions possess an antiderivative, i.e., there exists a function F with F △ = f , and in this case the delta integral is defined by where the integral on the right-hand side is the classical Riemann integral. If The delta integral has the following properties (see, e.g., [10]): (i) if f ∈ C prd and t ∈ T κ , then

The Weierstrass Necessary Condition
Let T be a bounded time scale. Throughout we let t 0 , t 1 ∈ T with t 0 < t 1 . For an interval [t 0 , t 1 ] ∩ T we simply write [t 0 , t 1 ]. The problem of the calculus of variations on time scales under consideration has the form over all x ∈ C 1 prd satisfying the boundary conditions A function x ∈ C 1 prd is said to be admissible if it satisfies conditions (3.2). Let us consider two norms in C 1 prd : where here and subsequently T denotes the set of points of [t 0 , t 1 ] κ where x △ (t) does not exist, and The norms · 0 and · 1 are called the strong and the weak norm, respectively. The strong and weak norms lead to the following definitions for local minimum: A weak minimum may not necessarily be a strong minimum: on the time scale T = { 1 n : n ∈ N} ∪ {0} (note that we need to add zero in order to have a closed set). Let us show thatx(t) = 0, 0 ≤ t ≤ 1 is a weak local minimum for (3.3). In the topology induced by · 1 consider the open ball of radius 1 centered atx, i.e., We use the notation B k r for the ball of radius r in norm · k , k = 1, 2. For every x ∈ B 1 1 (x) we have hence L[x] ≥ 0. This proves thatx is a weak local minimum for (3.3) since L[x] = 0. Now let us consider the function defined by Function x d is admissible and Therefore, for every δ > 0 there is a d such that We have , and x △ d (t) = 0 for all t = t 0 , σ(t 0 ). Hence, |x △ d (t)|, 0 ≤ t ≤ 1, can take arbitrary large values since µ(t) = t 2 1−t → 0 as t → 0. Note that for every δ > 0 we can choose d and t 0 such that x d ∈ B 0 δ (x) and d µ(σ(t0)) > 1. Finally, Therefore, the trajectoryx cannot be a strong minimum for (3.3).
From now on we assume that f : [t 0 , t 1 ] κ × R × R → R has partial continuous derivatives f x and f v , respectively with respect to the second and third variables, This function, called the Weierstrass excess function, is utilized in the following theorem: Theorem 3.3 (Weierstrass necessary optimality condition on time scales). Let T be a time scale, t 0 , t 1 ∈ T, t 0 < t 1 . Assume that the function f (t, x, r) in problem (3.1)-(3.2) satisfies the following condition: for each (t, x) ∈ [t 0 , t 1 ] κ × R, all r 1 , r 2 ∈ R and γ ∈ [0, 1]. Letx be a piecewise continuous function. Ifx is a strong local minimum for (3.1)-(3.2), then for all t ∈ [t 0 , t 1 ] κ and all q ∈ R, where we replacex △ (t) byx △ (t−) andx △ (t+) at finitely many points t wherex △ (t) does not exist.
Second, we suppose that a ∈ [t 0 , t 1 ] κ , a < t 1 , is a right-dense point and [a, b] ∩ T is an interval between two successive points wherex △ (t) does not exist. Then, there exists a sequence {ε k : k ∈ N} ⊂ [t 0 , t 1 ] with lim k→∞ ε k = a. Let τ be any number such that σ(τ ) ∈ [a, b) and q ∈ R. We define the function x : [t 0 , t 1 ] ∩ T → R as follows: Clearly, given δ > 0, for any q one can choose τ such that x −x 0 < δ.
so that, by Theorem 5.37 in [7] and Theorem 7.1 in [8], we obtain (3.7) Invoking the relation φ △1△2 = φ △2△1 (see Theorem 6.1 in [8]), integration by parts gives Sincex verifies the Euler-Lagrange equation (see [6]), we get On account of the above, from (3.6)-(3.7) we have To establish the condition (3.5) for all t ∈ [t 0 , t 1 ] κ , we consider the limit t → t 1 from left when t 1 is left-dense, and the limit t → t p from left and from right when t p ∈ T . Remark 3.5. Let T be a time scale with µ(t) depending on t and such that the time scale interval [t 0 , t 1 ] may be written as follows: [t 0 , t 1 ] = L ∪ U with µ(t) = 0 for all t ∈ L and µ(t) = 0 for all t ∈ U . An example of such time scale is the Cantor set [10]. Then, for t ∈ U the condition (3.4) is trivially satisfied, while for t ∈ L (3.4) is nothing more than convexity of f with respect to r.
Let now T = q N , q > 1. Ifx is a local minimum of the problem x(t 0 ) = α, x(t 1 ) = β, α, β ∈ R, and the function f (t, x, r) is convex with respect to r ∈ R for each (t, x) ∈ [t 0 , t 1 ) × R, then for all t ∈ [t 0 , t 1 ) and all p ∈ R.