Elza Farkhi

Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions

Ordinary differential and functional-differential inclusions with compact right-hand sides are considered. Stability theorems of Filippovs type in the convex and nonconvex case are proved under a one-sided Lipschitz condition, which extends the notions of Lipschitz continuity, dissipativity, and the uniform one-sided Lipschitz condition for set-valued mappings. The accuracy of approximation of the solution sets by means of the Euler discretization scheme for both types of inclusions is estimated.

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets

A normed and partially ordered vector space of so-called ‘directed sets’ is constructed, in which the convex cone of all nonempty convex compact sets in Rn is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n=1. The directed sets in Rn are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a ‘support’ function and directed ‘supporting faces’ of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the ‘support’ function and recursively on the directed ‘supporting faces’. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.

Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets

This paper is a continuation of the authors first paper (Set-Valued Anal.9 (2001), pp. 217–245), where the normed and partially ordered vector space of directed sets is constructed and the cone of all nonempty convex compact sets in Rn is embedded. A visualization of directed sets and of differences of convex compact sets is presented and its geometrical components and properties are studied. The three components of the visualization are compared with other known differences of convex compact sets.

Spline subdivision schemes for convex compact sets

The application of spline subdivision schemes to data consisting of convex compact sets, with addition replaced by Minkowski sums of sets, is investigated. These methods generate in the limit set-valued functions, which can be expressed explicitly in terms of linear combinations of integer shifts of B-splines with the initial data as coefficients. The subdivision techniques are used to conclude that these limit set-valued spline functions have shape-preserving properties similar to those of the usual spline functions. This extension of subdivision methods from the scalar setting to the set-valued case has application in the approximate reconstruction of 3-D bodies from finite collections of their parallel cross-sections.

Set-Valued Approximations with Minkowski Averages – Convergence and Convexification Rates

Abstract Three approximation processes for set-valued functions (multifunctions) with compact images in ℝ n are investigated. Each process generates a sequence of approximants, obtained as finite Minkowski averages (convex combinations) of given data of compact sets in ℝ n . The limit of the sequence exists and and is equal to the limit of the same process, starting from the convex hulls of the given data. The common phenomenon of convexification of the approximating sequence is investigated and rates of convergence are obtained. The main quantitative tool in our analysis is the Pythagorean type estimate of Cassels for the “inner radius” measure of nonconvexity of a compact set. In particular we prove the convexity of the images of the limit multifunction of set-valued spline subdivision schemes and provide error estimates for the approximation of set-valued integrals by Riemann sums of sets and for Bernstein type approximation to a set-valued function.

Fixed set iterations for relaxed Lipschitz multimaps

Abstract A dynamical system described by an autonomous differential inclusion with a right-hand side satisfying a relaxed Lipschitz condition, as well as its Euler approximations, are studied. We investigate the asymptotic properties of the solutions and of the attainable sets. It is shown that the system has a strongly flow invariant set, or a “fixed set”, that is, a set such that each trajectory starting from it does not leave it. This set is also an attractor, i.e., it attracts the continuous and the discrete Euler trajectories as the time tends to infinity. We give estimates of the rate of attraction. An algorithm for approximating the fixed set by the attainable sets of the discrete system is also presented.

On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions

The Mordukhovich subdifferential, being highly important in variational and nonsmooth analysis and optimization, often happens to be hard to calculate. We propose a method for computing the Mordukhovich subdifferential of differences of sublinear (DS) functions applying the directed subdifferential of differences of convex (DC) functions. We restrict ourselves to the two-dimensional case mainly for simplicity of the proofs and for the visualizations. The equivalence of the Mordukhovich symmetric subdifferential (the union of the corresponding subdifferential and superdifferential) to the Rubinov subdifferential (the visualization of the directed subdifferential) is established for DS functions in two dimensions. The Mordukhovich subdifferential and superdifferential are identified as parts of the Rubinov subdifferential. In addition, the Rubinov subdifferential may be constructed as the Mordukhovich one by Painleve–Kuratowski outer limits of Frechet subdifferentials. The results are applied to the case of DC functions. Examples illustrating the obtained results are presented. 2010 Mathematics Subject Classification. Primary 49J52; Secondary 26B25, 49J50, 90C26

Approximation of set-valued functions : adaptation of classical approximation operators

Scientific Background: On Functions with Values in Metric Spaces: Basic Notions, Approximation Operators On Sets: Sets, Set Operations and Parametrizations On Set-Valued Functions (SVFs): Representations and Regularity Approximation of SVFs: Methods Based on Canonical Representations Methods Based on Minkowski Convex Combinations Methods Based on the Metric Average Methods Based on Metric Linear Combinations Methods Based on Metric Selections Approximation of SVFs with Images in R: Regularity of the Boundaries of the Graph Multisegmental and Topological Representations Methods Based on Topological Representation.

Directed Subdifferentiable Functions and the Directed Subdifferential Without Delta-Convex Structure

We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the delta-convex structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions.

Discrete Approximations and Fixed Set Iterations in Banach Spaces

We study autonomous differential inclusions with right-hand sides satisfying a one-sided Lipschitz (OSL) condition in Banach spaces with uniformly convex duals. We first show that the solution set is closed and obtain estimates for Euler-type discrete approximations. We then use these results to derive an analogue of the exponential formula for the reachable set, as well as results regarding the existence and approximation of a strongly invariant attractor in the case of a negative OSL constant. As a by-product, conditions for controllability of the reverse-time system are obtained.