The Noether Principle of Optimal Control

2004

We obtain a generalization of Noether's invariance principle for optimal control problems with equality and inequality state-input constraints. The result relates the invariance properties of the problems with the existence of conserved quantities along the constrained Pontryagin extremals. A result of this kind was posed as an open question by Vladimir Tikhomirov, in 1986.

Set-Valued and Variational Analysis, 2019

This is a special issue honoring three distinguished and highly reputed researchers: Urszula Ledzewicz, Helmut Maurer and Heinz Schättler. Each of them, in their own way, has vastly contributed for the development of the field of Optimal Control theory and its application to real life problems. Helmut Maurer, Urszula Ledzewicz and Heinz Schättler have different backgrounds. Both Helmut and Heinz were born and educated in Germany. While Helmut remained in Germany throughout his career, Heinz headed to the US early in his life, where he got his PhD and where he has lived since then. Urszula Ledzewicz was born and educated in Poland. In the late 80's she moved to the US. Not surprisingly, her mathematical roots are in the Russian school. Helmut Maurer has always looked at any problem from three different sides: development of theory, applications and numerical methods. So, each piece of theory could not rest in peace before being tested in applications and some specific problems being solved numerically. He has offered us over the years important contributions to optimality conditions for constrained optimal control problems and to numerical methods for optimal control. Urszula Ledzewicz had first produced some impressive results on problems with mixed constraints in her early years while Heinz Schättler followed a Geometric Control approach to optimal control. They both turned to applications and, together, Urszula and Heinz broke new ground on optimal control of biomathematical problems, specially on the study of cancer treatment. They, many times together with Helmut, have shed a new light upon these problems.

In this paper we will propose a new method of approaching the problems of optimal control for a large class of dynamic systems. In comparison with the classical method of solving the problems of optimal control within the theory of dynamic systems, our method will be based exclusively on the Lagrangian variational calculus.This aspect has the advantage of excluding the complications resulted from the use of dual variational calculus and from the techniques developed by Carathéodory and Pointriaghin, which the old method depends of.

Economists’ Mathematical Manual, 1999

... Calculus of variations and optimal control theory. Post a Comment. CONTRIBUTORS: Author: Hestenes, Magnus R. (b. 1906, d. ----. ... VOLUME/EDITION: PAGES (INTRO/BODY): xii, 405 p. SUBJECT(S): Calculus of variations; Control theory. DISCIPLINE: No discipline assigned. ...

Proceedings of Institute of Mathematics of National …, 2004

The role of symmetry is well studied in physics and economics, where many great contributions have been made. With the help of Emmy Noether's celebrated theorems, a unified description of the subject can be given within the mathematical framework of the calculus of variations. It turns out that Noether's principle can be understood as a special application of the Euler-Lagrange differential equations. We claim that this modification of Noether's approach has the advantage to put the role of symmetry on the basis of the calculus of variations, and in a key position to give answers to some fundamental questions. We will illustrate our point with the interplay between the concept of invariance, the theory of optimality, Tonelli existence conditions, and the Lipschitzian regularity of minimizers for the autonomous basic problem of the calculus of variations. We then proceed to the general nonlinear situation, by introducing a concept of symmetry for the problems of optimal control, and extending the results of Emmy Noether to the more general framework of Pontryagin's maximum principle. With such tools, new results regarding Lipschitzian regularity of the minimizing trajectories for optimal control problems with nonlinear dynamics are obtained.

Econometrica, 1972

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Conservation laws, i.e. conserved quantities along Euler-Lagrange extremals, which are obtained on the basis of Noether's theorem, play an prominent role in mathematical analysis and physical applications. In this paper we present a general and constructive method to obtain conserved quantities along the Pontryagin extremals of optimal control problems, which are invariant under a family of transformations that explicitly change all (time, state, control) variables.

The Pontryagin's Maximum Principle allows, in most cases, the design of optimal controls of affine nonlinear control systems by considering the sign of a smooth function. There are cases, although, where this function vanishes on a whole time interval and the Pontryagin's Maximum Principle alone does not give enough information to design the control. In these cases one considers the time derivatives of this function until a k-order derivative that explicitly depends on the control variable. The number q=k/2 is called problem order and it is the same to all the extremals. The local order is a related concept used in literature, but depending on each particular extremal. The confusion between these two concepts led to misunderstandings in past works, where the problem order was assumed to be an integer number. In this work we prove that this is true if the control system has a single input but, in general, it is not true if the control system has a multiple input.

Journal of Optimization Theory and Applications, 1971

A sufficiency theorem is derived for optimal control of dynamical systems governed by ordinary differential equations. A simple example is given to illustrate the application of this theorem.

Applied Mathematics and Computation, 2011

... 1, ..., n) i = max j=1:ki i,j = max { i,1, ..., i,ki } , Email address: [email protected], [email protected] (Serkan Ilter) 1The ... Please be aware that although "Articles in Press" do not have all bibliographic details available yet, they can already be cited using the year of online ...

European Journal of Control, 2014

Some problems of Calculus of Variations do not have solutions in the class of classic continuous and smooth arcs. This suggests the need of a relaxation or extension of the problem ensuring the existence of a solution in some enlarged class of arcs. This work aims at the development of an extension for a more general optimal control problem with nonlinear control dynamics in which the control function takes values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the generalized controls, but related to discontinuous arcs. This leads to the notion of generalized impulsive control. The proposed extension links various approaches on the issue of extension found in the literature.