A unified approach to the calculus of variations on time scales

Abstract

Abstract:In this work we propose a new and more general approach to the calculus of variations on time scales that allows to obtain, as particular cases, both delta and nabla results. More precisely, we pose the problem of minimizing or maximizing the composition of delta and nabla integrals with Lagrangians that involve directional derivatives. Unified Euler-Lagrange necessary optimality conditions, as well as sufficient conditions under appropriate convexity assumptions, are proved. We illustrate presented results with simple examples.

Noether’s symmetry theorem for nabla problems of the calculus of variations

“…To the best of the authors knowledge there are now six techniques to obtain directly results for the nabla or delta calculus. These six approaches were introduced, respectively, in the following references (ordered by date, from the oldest approach to the most recent one): [2] (the alpha approach); [27] (the diamond-alpha approach); [3] (Aldwoah's or generalized time scales approach); [24] (the delta-nabla approach); [15] (Caputo's or duality approach); [19] (the directional approach). Paper [2] introduces the so-called alpha derivatives, where the σ operator in the definition of delta derivative (or ρ in the definition of nabla derivative) is substituted by a more general function α(·); paper [27] proposes a convex combination between delta and nabla derivatives: 1].…”

“…Such principle is illustrated in [15] by means of some examples, but is never proved (it is a principle, not a theorem). In [19] it is studied the problem of minimizing or maximizing the composition of delta and nabla integrals with Lagrangians that involve directional derivatives. In our paper we promote Caputo's technique, showing how her approach is simple and effective.…”

The contingent epiderivative and the calculus of variations on time scales

“…In this paper we claim that the contingent epiderivative, a concept introduced by Aubin with the help of his contingent derivative, is an important tool in the theory of time scales. To show this we propose a new approach to the calculus of variations on time scales, which allows to unify the three different approaches followed so far in the literature: the delta [9,16,23]; the nabla [2,3]; and the delta-nabla [17,27] approach. Such unification is obtained using, for the first time in the literature of time scales, the contingent epiderivative as the main differentiation tool.…”

“…Proofs of the following two theorems are done following the techniques in the proof of Theorem 34 by considering three cases: (i) u and w are both positive; (ii) u and w are both negative; (iii) u and w are of different signs. Indeed, in case (i) the isoperimetric problem (9)-(11) is reduced to the one studied in [16]; in case (ii) we can apply the results in [2] to obtain Theorem 36 and Theorem 37 in the case u, w < 0; when sign(uw) = −1 we need to use the necessary optimality conditions for the delta-nabla isoperimetric problems investigated in [17].…”

Delta-nabla optimal control problems

“…The following definition is motivated by the time scale Euler-Lagrange equations proved in [Girejko et al (2010)] and ].…”