Approximation of Reachable Sets using Optimal Control Algorithms

R. Baier, M. Gerdts, I. Xausa: Approximation of Reachable Sets using Optimal Control Algorithms
Numerical Algebra, Control and Optimization 3 (3), 519 - 548, 2013,  

DOI: 10.3934/naco.2013.3.519
MR Nummer: 3079193
Zentralblattnummer: 1271.49021
Keywords: reachable sets; discrete approximations; optimal control; direct discretization
Mathematics Subject Classification Code: 49J15 49M25 93B03 93C10 (90C30)


Abstract:

We investigate and analyze a computational method for the approximation of reachable sets for nonlinear dynamic systems. The method uses grids to cover the region of interest and the distance function to the reachable set evaluated at grid points. A convergence analysis is provided and shows the convergence of three different types of discrete set approximations to the reachable set. The distance functions can be computed numerically by suitable optimal control problems in combination with direct discretization techniques which allows adaptive calculations of reachable sets. Several numerical examples with nonconvex reachable sets are presented.

Contents:

1. Introduction
2. Proximal Normals and Inner/Outer Approximation of Sets
2.1 Set Representation Techniques
2.2 Inner/Outer Approximation of Sets
3. Convergence Analysis
3.1 Properties and Approximations of Reachable Sets
3.2 Discrete Approximation of Reachable Sets
4. Numerical Realization
4.1 DFOG Method
5. Numerical Examples
5.1 Kenderov's Example
5.2 Bilinear Example
5.3 Adaptive Version
5.4 Example from a Pursuit-Evasion Game
6. Outline

This article is the updated version of the technical report from October 17, 2011.

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