Robert Baier

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.

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L 1-norm and with order 1/2 w.r.t. the L 2-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.

Approximation of reachable sets by direct solution methods for optimal control problems

A numerical method for the approximation of reachable sets of linear control systems is discussed. The method is based on the formulation of suitable optimal control problems with varying objective function, whose discretization by Runge–Kutta methods leads to finite-dimensional convex optimization problems. It turns out that the order of approximation for the reachable set depends on the particular choice of the Runge–Kutta method in combination with the selection strategy used for control approximation. For an inappropriate combination, the expected order of convergence cannot be achieved in general. The method is illustrated by two test examples using different Runge–Kutta methods and selection strategies, in which the run times are analysed, the order of convergence is estimated numerically and compared with theoretical results in similar areas.

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.

Stability and Convergence of Euler's Method for State-Constrained Differential Inclusions

A discrete stability theorem for set-valued Eulers method with state constraints is proved. This theorem is combined with known stability results for differential inclusions with so-called smooth state constraints. As a consequence, order of convergence equal to 1 is proved for set-valued Eulers method, applied to state-constrained differential inclusions.

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

Selection strategies for set-valued runge-kutta methods

A general framework for proving an order of convergence for set-valued Runge Kutta methods is given in the case of linear differential inclusions, if the attainable set at a given time should be approximated. The set-valued method is interpreted as a (set-valued) quadrature method with disturbed values for the fundamental solution at the nodes of the quadrature method. If the precision of the quadrature method and the order of the disturbances fit together, then an overall order of convergence could be guaranteed. The results are applied to modified Euler method to emphasize the dependence on a suitable selection strategy (one strategy leads to an order breakdown).

Discrete Approximation of Impulsive Differential Inclusions

The article deals with the approximation of the solution set and the reachable sets of an impulsive differential inclusion with variable times of impulses. It is strongly connected to [11] and is its continuation. We achieve order of convergence 1 for the Euler approximation under Lipschitz assumptions on the set-valued right-hand side and on the functions describing the jump surfaces and jumps themselves. Another criterion prevents the beating phenomena and generalizes available conditions. Several test examples illustrate the conditions and the practical evaluation of the jump conditions.

Set-valued Hermite interpolation

Abstract The problem of interpolating a set-valued function with convex images is addressed by means of directed sets. A directed set will be visualised as a usually non-convex set in R n consisting of three parts together with its normal directions: the convex, the concave and the mixed-type part. In the Banach space of the directed sets, a mapping resembling the Kergin map is established. The interpolating property and error estimates similar to the point-wise case are then shown; the representation of the interpolant through means of divided differences is given. A comparison to other set-valued approaches is presented. The method developed within the article is extended to the scope of the Hermite interpolation by using the derivative notion in the Banach space of directed sets. Finally, a numerical analysis of the explained technique corroborates the theoretical results.

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.