Gauge Symmetries and Noether Currents in Optimal Control

We extend the second Noether theorem to optimal control problems which are invariant under symmetries depending upon k arbitrary functions of the independent variable and their derivatives up to some order m. As far as we consider a semi-invariance notion, and the transformation group may also depend on the control variables, the result is new even in the classical context of the calculus of variations.Comment: Partially presented at the 5th Portuguese Conference on Automatic Control (Controlo 2002), Aveiro, Portugal, September 5-7, 2002. Accepted for publication in Applied Mathematics E-Notes, Volume 3. See http://www.mat.ua.pt/delfim for other work

Caratheodory-Equivalence, Noether Theorems, and Tonelli Full-Regularity in the Calculus of Variations and Optimal Control

We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are connected to: i) the Caratheodory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the theorems of Noether; iii) regularity conditions for the minimizers and the work pioneered by Tonelli.Comment: 24 pages, Submitted for publication in a Special Issue of the J. of Mathematical Science

Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems

We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions we can assure optimal controls are bounded? This question is related to the one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial for closing the gap between the conditions arising in the existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the later problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover optimal control problems with general nonlinear dynamics.Comment: This research was partially presented, as an oral communication, at the international conference EQUADIFF 10, Prague, August 27-31, 2001. Accepted for publication in the journal Mathematics of Control, Signals, and Systems (MCSS). See http://www.mat.ua.pt/delfim for other work

Weak Conservation Laws for Minimizers which are not Pontryagin Extremals

We prove a Noether-type symmetry theorem for invariant optimal control problems with unrestricted controls. The result establishes weak conservation laws along all the minimizers of the problems, including those minimizers which do not satisfy the Pontryagin Maximum Principle.Comment: Accepted for presentation (Paper No: 113) at the 2nd International Conference "Physics and Control" (PhysCon 2005), August 24-26, 2005, Saint Petersburg, Russia. To appear in the respective Conference Proceeding

A survey on fractional variational calculus

Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental, higher-order, and isoperimetric problems, and compute approximated solutions based on truncated Gr\"{u}nwald--Letnikov approximations of Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form is in 'Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory', De Gruyter. Submitted 29-March-2018; accepted, after a revision, 13-June-201

Optimal Control and Sensitivity Analysis of a Fractional Order TB Model

A Caputo fractional-order mathematical model for the transmission dynamics of tuberculosis (TB) was recently proposed in [Math. Model. Nat. Phenom. 13 (2018), no. 1, Art. 9]. Here, a sensitivity analysis of that model is done, showing the importance of accuracy of parameter values. A fractional optimal control (FOC) problem is then formulated and solved, with the rate of treatment as the control variable. Finally, a cost-effectiveness analysis is performed to assess the cost and the effectiveness of the control measures during the intervention, showing in which conditions FOC is useful with respect to classical (integer-order) optimal control.Comment: This is a preprint of a paper whose final and definite form is with 'Statistics Opt. Inform. Comput.', Vol. 7, No 2 (2019). See [http://www.IAPress.org]. Submitted 09/Sept/2018; Revised 10/Dec/2018; Accepted 11/Dec/2018. arXiv admin note: text overlap with arXiv:1801.09634, arXiv:1810.0690