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

Scattering on Dislocations and Cosmic Strings in the Geometric Theory of Defects

We consider scattering of elastic waves on parallel wedge dislocations in the geometric theory of defects or, equivalently, scattering of point particles and light rays on cosmic strings. Dislocations are described as torsion singularities located on parallel lines, and trajectories of phonons are assumed to be the corresponding extremals. Extremals are found for arbitrary distribution of the dislocations in the monopole, dipole, and quadrupole approximation and the scattering angle is obtained. Examples of continuous distribution of wedge and edge dislocations are considered. We have found that for deficit angles close to -2\pi a star located behind a cosmic string may have any even number of images, 2,4,6,... The close relationship between dislocations and conformal maps is elucidated in detail.Comment: 30 pages, 14 figures, minor change

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