Optimal Control with Time Delays via the Penalty Method

Abstract

We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.

Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

“…Many studies relating to controller design, stability studies, neural models, and fractional problems have addressed multi-time-delay [41][42][43][44][45]. Benharrat and Torres [46] studied variation problems of the optimal control problem via the penalty method by considering multi-time-delay. The variational principles on mechanical systems and the corresponding symmetry theory are still poorly studied in terms of taking into account multi-time-delay.…”

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

“…Motivated by the important applications of Noether's second theorem [20] and the applicability of higher-order dynamic systems with time delays in modeling real-life phenomena [4,7,29], as well as the importance of variational problems of Herglotz [14,16], our goal in this paper is to study generalized variational problems that are invariant under a certain group of transformations that depends on arbitrary functions and their derivatives up to some order, and deduce expressions for Noether currents, that is, expressions that are constant in time along the extremals.…”