Higher-Order Variational Problems of Herglotz Type

Abstract

We obtain a generalized Euler-Lagrange differential equation and transversality optimality conditions for Herglotz-type higher-order variational problems. Illustrative examples of the new results are given.

Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres

“…As it can be easily checked, this corresponds exactly to the Euler-Lagrange equation for the second-order Herglotz problem in Euclidean spaces studied in [34].…”

Fractional Herglotz variational principles with generalized Caputo derivatives

“…As an application, a fractional mechanical system is analyzed with a fractionally generalized velocity that reproduces, for α = 1, the standard Lagrangian of a harmonic oscillator with exponential damping, which also contains the non-damped conservative oscillator. The research here initiated can now be enriched in different directions, by trying to bring to the fractional setting the recent results [28,29,30,31,32] of Santos et al on Herglotz variational problems.…”

“…The readers interested in the discrete fractional calculus of variations are refereed to the pioneer work of Bastos et al [8,9]. Here we are interested in the generalized continuous calculus of variations introduced by Herglotz. The generalized variational principle firstly proposed by Gustav Herglotz in 1930 [14] gives a variational principle description of non-conservative systems even when the Lagrangian is autonomous [28,30]. It is essentially based on the following problem: find the trajectories x(t), satisfying given boundary conditions, that extremize (minimize or maximize) the terminal value z(b) of the functional z that satisfies the differential equatioṅ…”

An Optimal Control Approach to Herglotz Variational Problems

“…It consists in the determination of trajectories x(·) and corresponding trajectories z(·) that extremize (maximize or minimize) the value z(b), where L ∈ C 1 ([a, b] × R 2n × R; R). While in [3,4,6] the admissible functions are x(·) ∈ C 2 ([a, b]; R n ) and z(·) ∈ C 1 ([a, b]; R), here we consider (P H ) in the wider class of functions x(·) ∈ P C 1 ([a, b]; R n ) and z(·) ∈ P C 1 ([a, b]; R).…”

“…Note that the classical problem of the calculus of variations (1) is a particular case of problem (P ) with φ(x) ≡ 0, g(t, x, u) = u and Ω = R n . In this work we show how the results on Herglotz's problem of the calculus of variations (P H ) obtained in [2,6] can be generalized by using the theory of optimal control. The main idea is simple and consists in rewriting the generalized variational problem of Herglotz (P H ) as a standard optimal control problem (P ), and then to apply available results of optimal control theory.…”