Noether currents for higher-order variational problems of Herglotz type with time delay

Abstract

We study, from an optimal control perspective, Noether currents for higher-order problems of Herglotz type with time delay. Main result provides new Noether currents for such generalized variational problems, which are particularly useful in the search of extremals. The proof is based on the idea of rewriting the higher-order delayed generalized variational problem as a first-order optimal control problem without time delays.

Optimality conditions involving the Mittag–Leffler tempered fractional derivative

“…The principle of Herglotz was presented in 1930 [22], but only more recently with the works of Georgieva and coauthors [19,20,21] it has gained the attention of more researchers (e.g. [1,4,36,37,38]). The classical Herglotz problem states as follows: determine the trajectories x ∈ C 1 [a, b] and the corresponding function z ∈ C 1 [a, b] such that z(b) attains a minimum value, where the pair (x, z) satisfies the ODE…”

Scale Symmetry and Friction

“…Let us now consider the generalization of these systems to include the action, S, as part of the Lagrangian L H (q,q, S); hence, L H is a function on TM × R. These were first considered by Herglotz as a way to introduce non-conservative terms and have found a variety of applications [25][26][27][28][29][30] across applied mathematics. Hence, we shall call these 'Herglotz Lagrangians'.…”

Conservation laws for Birkhoffian systems of Herglotz type

“…Theorem 5 For the constrained Birkhoffian system ( 7), (9) of Herglotz type, if a conserved quantity of the system is known, then the infinitesimal generators F µ , f and the gauge function G can be found by formulae (38), (39) and the restriction Eq. (30).…”

“…In recent years, the Herglotz variational principle and its symmetries have been applied in finite and infinite dimensional non-conservative dynamic systems, quantum systems, thermodynamics, optimal control theory, and other fields. [24][25][26][27][28][29][30][31][32][33] In Ref. [34], the simple and physically meaningful Lagrangians of Herglotz type were constructed, which describe a wide range of non-conservative classical and quantum systems, for example, vibrating string under viscous forces, non-conservative electromagnetic theory, non-conservative Schrödinger equation, non-conservative Klein-Gordon equation, etc.…”