Optimal Control Theory: Introduction to the Special Issue

Econometrica, 1972

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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.

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.

2019

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. An optimal control is an extension of the calculus of variations. It is a mathematical optimization method for deriving control policies. The calculus of variations is concerned with the extrema of functionals. The different approaches tried out in its solution may be considered, in a more or less direct way, as the starting point for new theories. While the true “mathematical” demonstration involves what we now call the calculus of variations, a theory for which Euler and then Lagrange established the foundations, the solution which Johann Bernoulli originally produced, obtained with the help analogy with the law of refraction on optics, was empirical. A similar analogy between optics and mechanics reappears when Hamilton applied the principle of least action in mechanics which Maupertuis justified in the first instance, on the basis of the laws o...

HAL (Le Centre pour la Communication Scientifique Directe), 2021

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This article deals with the main points of numerical solution of optimal control problems basing on the method of variations. We give a description of an algorithm of the method of variations a numerical solution of problems of optimal control. Finally, numerical examples are included to demonstrate efficiency, simplicity and high accuracy of the proposed method.

Page 1. The Noether Principle of Optimal Control Delfim FM Torres [email protected] Department of Mathematics University of Aveiro 3810–193 Aveiro, Portugal http://www.mat.ua.pt/delfim Page 2.

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.

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.

Journal of Differential Equations, 2012

The paper is concerned with problems of optimal feedback control with "nonclassical" dynamicsẋ = f (t, x, u, Du), where the evolution of the state x depends also on the Jacobian matrix Du = (∂u i /∂x j) of the feedback control function u = u(t, x). Given a probability measure µ on the set of initial states, we seek feedback controls u(•) which minimize the expected value of a cost functional. After introducing a basic framework for the study of these problems, this paper focuses on three main issues: (i) necessary conditions for optimality, (ii) equivalence with a relaxed feedback control problem in standard form, and (iii) dependence of the expected minimum cost on the probability measure µ.

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.