Nonshifted calculus of variations on time scales with ∇-differentiable σ

Abstract

In calculus of variations on general time scales, an Euler-Lagrange equation of integral form is usually derived in order to characterize the critical points of nonshifted Lagrangian functionals, see e.g. [R.A.C. Ferreira and co-authors, Optimality conditions for the calculus of variations with higher-order delta derivatives, Appl. Math. Lett., 2011]. In this paper, we prove that the ∇-differentiability of the forward jump operator σ is a sharp assumption on the time scale in order to ∇-differentiate this integral Euler-Lagrange equation. This procedure leads to an Euler-Lagrange equation of differential form. Furthermore, from this differential form, we prove a Noether-type theorem providing an explicit constant of motion for Euler-Lagrange equations admitting a symmetry.

Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales

“…This provides an illustration that the time scale theory allows to close the gap between continuous and discrete analyses, and this is possible in any mathematical domain in which time scale calculus can be involved. Another example is provided in optimization with the calculus of variations on time scales, initiated in [8], and well-studied in the literature (see, e.g., [7,15,43,46]). On the other hand, in [44,45], the authors establish a weak version of the PMP (with the nonpositive Hamiltonian gradient condition) for (permanent) control systems defined on general time scales.…”

Optimal sampled-data control, and generalizations on time scales

“…The theory of the calculus of variations on time scales, initiated in [8], has been well studied in the existing literature (see, e.g., [7,9,17,28,35,38]). In [36,37], the authors establish a weak version of the PMP (with a nonpositive gradient condition) for control systems defined on general time scales.…”

Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

“…However, the research is mainly limited to the following: (1) conservative system, (2) Noether symmetry, and (3) Noether-type conservation laws. Furthermore, according to Bourdin's study [45], the results of the nonshifted case at the discrete level maintain the variational structure and related properties, although so far there have been few studies on the time-scale nonshifted variational problem. This paper will focus on exploring Mei symmetry for time-scale nonshifted general holonomic systems and nonholonomic systems under the Hamiltonian framework, prove Mei symmetry theorems, and derive Mei conservation laws.…”