# Viability Kernels and Control Sets

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 5, page 175-185
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topSzolnoki, Dietmar. "Viability Kernels and Control Sets." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 175-185. <http://eudml.org/doc/197374>.

@article{Szolnoki2010,

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.
},

author = {Szolnoki, Dietmar},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Control affine system;
viability kernel; reachable set; control set; chain control set;
control flow.; control affine system; control flow; viability kernels},

language = {eng},

month = {3},

pages = {175-185},

publisher = {EDP Sciences},

title = {Viability Kernels and Control Sets},

url = {http://eudml.org/doc/197374},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Szolnoki, Dietmar

TI - Viability Kernels and Control Sets

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 5

SP - 175

EP - 185

AB -
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.

LA - eng

KW - Control affine system;
viability kernel; reachable set; control set; chain control set;
control flow.; control affine system; control flow; viability kernels

UR - http://eudml.org/doc/197374

ER -

## References

top- J.-P. AUBIN, Viability Theory. Birkhäuser (1991).
- F. COLONIUS AND W. KLIEMANN, Infinite 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 Equations5 (1993) 469-494.
- height 2pt depth -1.6pt width 23pt, The Dynamics of Control. Birkhäuser (2000) to appear.
- M. DELLNITZ AND A. HOHMANN, 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).
- P. SAINT-PIERRE, Approximation 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).
- A. UPPAL, W.H. RAY AND A.B. POORE, On the dynamic behavior of continuous stirred tank reactors. Chem. Engrg. Sci.19 (1974) 967-985.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.