Discrete approximations of differential inclusions

F. Lempio, V. M. Veliov: Discrete approximations of differential inclusions
in: Department of Mathematics, University of Bayreuth (eds.)
Bayreuther Mathematische Schriften 54, Bayreuth, Germany, 1998, 149 - 232

Smart-Link: http://bms.math.uni-bayreuth.de/all_books.html#vol_54
ISBN/ISSN/ISMV Nummer: 0172-1062
MR Nummer: 1643941
Zentralblattnummer: 0922.65059
Keywords: differential inclusions; Runge-Kutta methods; discrete approximations; reachable sets; set-valued numerical analysis
Mathematics Subject Classification Code: 49M25 (34A60 49D25 65L05)


This paper is devoted to discretization methods for initial value problems for differential inclusions. Main emphasis is laid on set-valued analogues of Runge-Kutta methods.

The following questions are studied:

  1. Does every discrete trajectory approximate some exact solution, and with what accuracy?
  2. Is every exact solution approximated, and with what accuracy, by some discrete trajectory?
  3. What is the error between the discrete reachable set and the exact reachable set with respect to Hausdorff distance?

The material is presented as a survey, containing simultaneously a series of new results with complete proofs.


1. Introduction and Preliminaries
2. Approximation of Some Trajectory
3. Approximation of All Trajectories
4. Hausdorff Approximation of the Reachable Set
5. Connection with Optimal Control

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