Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-19T23:57:48.615Z Has data issue: false hasContentIssue false
  • English
  • Français

Viability Kernels and Control Sets

Published online by Cambridge University Press:  15 August 2002

Dietmar Szolnoki
Dietmar Szolnoki*
Affiliation:
Universität Augsburg, Institut für Mathematik, Universitätsstraße 14, 86135 Augsburg, Germany; [email protected].
Get access

Abstract

This paper analyzes the relation of viability kernels and control sets of control affine systems. A viability kernel describes the largest closed viability domain contained in some closed subset Q of the state space. On the other hand, control sets are maximal regions of the state space where approximate controllability holds. It turns out that the viability kernel of Q can be represented by the union of domains of attraction of chain control sets, defined relative to the given set Q. In particular, with this result control sets and their domains of attraction can be computed using techniques for the computation of attractors and viability kernels.

Keywords

Control affine systemviability kernelreachable setcontrol setchain control setcontrol flow.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 175 - 185
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.-P. AUBIN, Viability Theory. Birkhäuser (1991).
Infinite, F. COLONIUS AND W. KLIEMANN time optimal control and periodicity. Appl. Math. Optim. 20 (1989) 113-130.
height 2pt depth -1.6pt width 23pt, Some, aspects of control systems as dynamical systems. J. Dynam. Differential Equations 5 (1993) 469-494.
height 2pt depth -1.6pt width 23pt, The Dynamics of Control. Birkhäuser (2000) to appear.
HOHMANN, M. DELLNITZ AND A., A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75 (1997) 293-317.
G. HÄCKL, Reachable Sets, Control Sets and Their Computation. Dissertation, Universität Augsburg, ``Augsburger Mathematische Schriften Band 7" (1996).
W. KLIEMANN, Qualitative Theorie Nichtlinearer Stochastischer Systeme. Dissertation, Universität Bremen (1980).
H. NIJMEIJER AND A.J. VAN DER SCHAFT, Nonlinear Dynamical Control Systems. Springer-Verlag (1990).
Approximation, P. SAINT-PIERRE of the viability kernel. Appl. Math. Optim. 29 (1994) 187-209.
height 2pt depth -1.6pt width 23pt, Set-valued numerical analysis for optimal control and differential games (1998) to appear.
D. SZOLNOKI, Berechnung von Viabilitätskernen. Diplomarbeit, Institut für Mathematik, Universität Augsburg, Augsburg (1997).
UPPAL, A., On, W.H. RAY AND A.B. POORE the dynamic behavior of continuous stirred tank reactors. Chem. Engrg. Sci. 19 (1974) 967-985. CrossRef