Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids

Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold (M,g). In this paper we study the restrictions on the topology and geometry of the fibres (the level sets) of the solutions f to (P1). We give a technique based on certain remarkable property of the fibres (the analytic representation property) for going from the initial PDE to a global analytical characterization of the fibres (the equilibrium partition condition). We study this analytical characterization and obtain several topological and geometrical properties that the fibres of the solutions must possess, depending on the topology of M and the metric tensor g. We apply these results to the classical problem in physics of classifying the equilibrium shapes of both Newtonian and relativistic static self-gravitating fluids. We also suggest a relationship with the isometries of a Riemannian manifold.Comment: 36 pages. In this new version the analytic representation hypothesis is proved. Please address all correspondence to D. Peralta-Sala

New Theories of Set-Valued Differentials and New Versions of the Maximum Principle of Optimal Control Theory

The purpose of this note is to announce a new theory of generalized differentials---the "generalized di#erential quotients," abbr. GDQs--- which has good open mapping properties, and to use this theory to state---in Theorem 9.4--- a version of the maximum principle for hybrid optimal control problems under weak regularity conditions. For single-valued maps, our GDQ theory essentially coincides with the one proposed by H. Halkin in [4], but GDQ theory applies as well to multivalued maps, thus making it possible to deal with non-Lipschitz vector fields, whose flow maps are in general set-valued. The results presented here are much weaker than what can actually be proved by our methods. More general versions, involving systems of di#erential inclusions, are discussed in a detailed paper currently in preparation. The GDQ concept contains several other notions of generalized differential, but does not include some important theories such as J. Warga's "derivate containe

Two New Methods for Motion Planning for Controllable Systems without Drift

Introduction The purpose of this note is to describe two new general procedures for constructing (open-loop) control functions for systems without drift, i.e. for solving the Motion Planning Problem (MPP) for nonholonomic systems without drift. Following the early work of Brockett and Sastry, cf. [3], [11], many methods have been proposed for solving this problem, e.g. by Sastry, Hauser, Murray and Li. In this note we outline some results, described in more detail in [9], [10], [16], [17], [18], that provide a new approach to the problem. Consider a control system \Sigma : x = m X i=1 u i f i (x); (1) where f 1 ; f 2 ; : : : ; f m are smooth vector fields in IR n . An admissible control for System (1) is a Lebesgue integrable function

Warga Derivate Containers and Other Generalized Differentials

This is the first of two papers devoted to recent ideas on the theory of generalized differentials with good open mapping properties. Here we will discuss "generalized differentiation theories" (abbr. GDTs), with special emphasis on the series of developments initiated by Jack Warga's pioneering work on derivate containers. In the second paper, we will focus on the most recent theory, of "path-integral generalized differentials," and prove that it has the crucial properties required for a version of the Pontryagin Maximum Principle (abbr. PMP) to exist, namely, the chain rule and the directional open mapping property. Our work continues the study of general smooth, nonsmooth, high-order, and hybrid versions of the PMP for finite-dimensional deterministic optimal control problems without state space constraints by means of a method developed by us in recent years. As explained in [11, 12, 13, 14], such versions can be derived in a unified way, by using a modified version of..

Shortest Paths in R³ with a Prescribed Curvature Bound

this paper, the abbreviation "PAL" stands for "parametrized by arc-length." "Circle" means "arc of a PAL circle of radius one," and "segment" means "PAL straight line segment. Circles satisfy the equation

Nonlinear Output Feedback Design for Linear Systems With Saturating Controls

This paper shows the existence of (nonlinear) smooth dynamic feedback stabilizers for linear time invariant systems under input constraints, assuming only that open-loop asymptotic controllability and detectability hold. 1 Introduction The study of actuator saturation in linear control design has a long history; see for instance [1], in particular Chapter 12 on dual-mode regulators, and the references given there. The search for controllers of systems subject to such saturation can be seen as a problem in nonlinear control, and that is the point of view taken here. In particular, we look at questions of stabilization, an area that has witnessed a large amount of activity during the last few years (see for instance [6] for a survey and many bibliographical references). In this paper we provide a general result on smooth stabilizability under minimal (and clearly necessary) hypotheses. The systems that we deal with have the form x = Ax + B `(u) (SYS) where A and B are n \Theta n and n ..