On symmetries in optimal control

Journal of Geometry and Physics, 2001

Symmetries in vakonomic dynamics are discussed. Appropriate notions are introduced and their relationship with previous work on symmetries of singular Lagrangian systems is shown. Some Noether-type theorems are obtained. The results are applied to a class of general optimal control problems and to kinematic locomotion systems.

Games

Optimal control theory is a modern extension of the classical calculus of variations [...]

2007

arXiv preprint math/0604072, 2006

We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether's first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for the sub-Riemannian nilpotent problem (2, 3, 5, 8).

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.

2012

An interesting family of geometric integrators can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators, being one of their main properties the preservation of geometric features as the symplecticity, momentum preservation and good behavior of the energy. We construct variational integrators for higher-order mechanical systems on trivial principal bundles and their extension for higher-order constrained systems and we devote special attention to the particular case of underactuated mechanical systems

Applied Mathematics E-Notes, 2003

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.

Portugaliæ Mathematica, 2004

We study in optimal control the important relation between invariance of the problem under a family of transformations, and the existence of preserved quantities along the Pontryagin extremals. Several extensions of Noether theorem are provided, in the direction which enlarges the scope of its application. We formulate a more general version of Noether's theorem for optimal control problems, which incorporates the possibility to consider a family of transformations depending on several parameters and, what is more important, to deal with quasi-invariant and not necessarily invariant optimal control problems. We trust that this latter extension provides new possibilities and we illustrate it with several examples, not covered by the previous known optimal control versions of Noether's theorem.

Econometrica, 1972

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2003

Control theory is a young branch of mathematics that has developed mostly in the realm of engineering problems. It is splitted in two major branches; control theory of problems described by partial difierential equa-tions where control are exercized either by boundary terms and/or inhomoge-neous terms and where the objective functionals are mostly quadratic forms; and control theory of problems described by parameter dependent ordinary difierential equations. In this case it is more frequent to deal with non-linear systems and non-quadratic objective functionals [49]. In spite that control theory can be consider part of the general theory of difierential equations, the problems that inspires it and some of the results obtained so far, have configured a theory with a strong and definite personality that is already of-fering interesting returns to its ancestors. For instance, the geometrization of nonlinear a卤ne-input control theory problems by introducing Lie-geometrical methods into...

Arxiv preprint arXiv: …, 2010

Abstract: We discuss the use of Dirac structures to obtain a better understanding of the geometry of a class of optimal control problems and their reduction by symmetries. In particular we will show how to extend the reduction of Dirac structures recently proposed ...

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.

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.

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.

1982

A general formalism is introduced for the optimal control problem on manifolds. It is based on a general formulation of Lagrange's multiplier theorem and recent definitions of nonlinear control systems. It is shown that we can give Pontryagin's maximum principle in this formalism. We expect that the problem formulation given in this paper is particularly suitable for application of modern results about controllability etc. in nonlinear control systems.