InforLorV4, Main, Curation, bibRecord, 009206

Deciding stability and mortality of piecewise affine dynamical systems

Identifieur interne : 009206 ( Main/Curation ); précédent : 009205; suivant : 009207

Deciding stability and mortality of piecewise affine dynamical systems

Auteurs : Vincent D. Blondel [Belgique] ; Olivier Bournez [France] ; Pascal Koiran [France] ; Christos H. Papadimitriou [États-Unis] ; John N. Tsitsiklis [États-Unis]

Source :

Theoretical Computer Science [ 0304-3975 ] ; 2001.

RBID : ISTEX:7413000D1133CDB096E3520BEE1F700C7AD0FEE0

Descripteurs français

Pascal (Inist)
- Décidabilité, Modèle 1 dimension, Modèle 2 dimensions, Mortalité, Stabilité, Système affine, Système discret, Système dynamique, Système hybride, Système linéaire, Système linéaire par morceau.
Wicri :
- topic : Mortalité.

English descriptors

Abstract

Abstract: In this paper we study problems such as: given a discrete time dynamical system of the form x(t+1)=f(x(t)) where f:Rn→Rn is a piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem that this Attractivity Problem is undecidable as soon as n⩾2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (do all trajectories go through 0?). We then show that Attractivity and Stability become decidable in dimension 1 for continuous functions.

Url:

DOI: 10.1016/S0304-3975(00)00399-6

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Le document en format XML

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<front><div type="abstract" xml:lang="en">Abstract: In this paper we study problems such as: given a discrete time dynamical system of the form x(t+1)=f(x(t)) where f:Rn→Rn is a piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem that this Attractivity Problem is undecidable as soon as n⩾2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (do all trajectories go through 0?). We then show that Attractivity and Stability become decidable in dimension 1 for continuous functions.</div>
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<author><name sortKey="Tsitsiklis, John N" sort="Tsitsiklis, John N" uniqKey="Tsitsiklis J" first="John N." last="Tsitsiklis">John N. Tsitsiklis</name>
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<front><div type="abstract" xml:lang="en">We show that several global properties (attractivity, global asymptotic stability and mortality) of discrete time dynamical systems defined by iteration of piecewise-affine maps are undecidable. Such results had been known only for local properties (e.g., point-to-point reachability). These three properties are undecidable in dimension at least two, but turn out to be decidable in one dimension for continuous maps. This gives a partial answer to a question of Sontag on the decidability of the stability of saturated linear dynamical systems.</div>
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<sourceDesc><biblStruct><analytic><title xml:lang="en" level="a">Deciding stability and mortality of piecewise affine dynamical systems</title>
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<s3>BEL</s3>
<sZ>1 aut.</sZ>
</inist:fA14>
<country>Belgique</country>
<placeName><region type="land" nuts="2">Vienne (Autriche)</region>
<settlement type="city">Vienne (Autriche)</settlement>
</placeName>
<orgName type="university">Université catholique de Louvain</orgName>
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<author><name sortKey="Bournez, Olivier" sort="Bournez, Olivier" uniqKey="Bournez O" first="Olivier" last="Bournez">Olivier Bournez</name>
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<author><name sortKey="Papadimitriou, Christos H" sort="Papadimitriou, Christos H" uniqKey="Papadimitriou C" first="Christos H." last="Papadimitriou">Christos H. Papadimitriou</name>
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<title level="j" type="abbreviated">Theor. comput. sci.</title>
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<profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Affine system</term>
<term>Decidability</term>
<term>Discrete system</term>
<term>Dynamical system</term>
<term>Hybrid system</term>
<term>Linear system</term>
<term>Mortality</term>
<term>One dimensional model</term>
<term>Picewise linear system</term>
<term>Stability</term>
<term>Two dimensional model</term>
</keywords>
<keywords scheme="Pascal" xml:lang="fr"><term>Système dynamique</term>
<term>Stabilité</term>
<term>Décidabilité</term>
<term>Mortalité</term>
<term>Système linéaire</term>
<term>Système hybride</term>
<term>Système discret</term>
<term>Modèle 1 dimension</term>
<term>Modèle 2 dimensions</term>
<term>Système affine</term>
<term>Système linéaire par morceau</term>
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<keywords scheme="Wicri" type="topic" xml:lang="fr"><term>Mortalité</term>
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<front><div type="abstract" xml:lang="en">In this paper we study problems such as: given a discrete time dynamical system of the form x(t + 1) = f(x(t)) where f : R<sup>n</sup>
 → R<sup>n</sup>
 is a piecewise affine function, decide whether all trajectories converge to 0, We show in our main theorem that this Attractivity Problem is undecidable as soon as n ≥ 2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (do all trajectories go through 0?). We then show that Attractivity and Stability become decidable in dimension 1 for continuous functions.</div>
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<sourceDesc><biblStruct><analytic><title level="a">Deciding stability and mortality of piecewise affine dynamical systems</title>
<author><name sortKey="Blondel, Vincent D" sort="Blondel, Vincent D" uniqKey="Blondel V" first="Vincent D." last="Blondel">Vincent D. Blondel</name>
<affiliation wicri:level="4"><country xml:lang="fr">Belgique</country>
<wicri:regionArea>CESAME, Université Catholique de Louvain, 4 Av. Georges Lemaitre, B-1348 Louvain-la-Neuve</wicri:regionArea>
<orgName type="university">Université catholique de Louvain</orgName>
<placeName><settlement type="city">Louvain-la-Neuve</settlement>
<region type="region" nuts="1">Région wallonne</region>
<region type="province" nuts="1">Province du Brabant wallon</region>
</placeName>
</affiliation>
<affiliation></affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">Belgique</country>
</affiliation>
</author>
<author><name sortKey="Bournez, Olivier" sort="Bournez, Olivier" uniqKey="Bournez O" first="Olivier" last="Bournez">Olivier Bournez</name>
<affiliation wicri:level="3"><country xml:lang="fr">France</country>
<wicri:regionArea>LORIA, Campus Scientifique BP 239, F-54506 Vandoeuvre-les-Nancy</wicri:regionArea>
<placeName><region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
<settlement type="city">Vandœuvre-lès-Nancy</settlement>
</placeName>
</affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">France</country>
</affiliation>
</author>
<author><name sortKey="Koiran, Pascal" sort="Koiran, Pascal" uniqKey="Koiran P" first="Pascal" last="Koiran">Pascal Koiran</name>
<affiliation wicri:level="3"><country xml:lang="fr">France</country>
<wicri:regionArea>LIP, ENS Lyon, 46 allée d'Italie, F-69364 Lyon Cedex 07</wicri:regionArea>
<placeName><region type="region" nuts="2">Auvergne-Rhône-Alpes</region>
<region type="old region" nuts="2">Rhône-Alpes</region>
<settlement type="city">Lyon</settlement>
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<affiliation wicri:level="1"><country wicri:rule="url">France</country>
</affiliation>
</author>
<author><name sortKey="Papadimitriou, Christos H" sort="Papadimitriou, Christos H" uniqKey="Papadimitriou C" first="Christos H." last="Papadimitriou">Christos H. Papadimitriou</name>
<affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country>
<wicri:regionArea>Department of Computer Sciences, University of California, Berkeley, CA 94720</wicri:regionArea>
<placeName><region type="state">Californie</region>
</placeName>
</affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">États-Unis</country>
</affiliation>
</author>
<author><name sortKey="Tsitsiklis, John N" sort="Tsitsiklis, John N" uniqKey="Tsitsiklis J" first="John N." last="Tsitsiklis">John N. Tsitsiklis</name>
<affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country>
<wicri:regionArea>LIDS, MIT, Cambridge, MA 02139</wicri:regionArea>
<placeName><region type="state">Massachusetts</region>
</placeName>
</affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">États-Unis</country>
</affiliation>
</author>
</analytic>
<monogr></monogr>
<series><title level="j">Theoretical Computer Science</title>
<title level="j" type="abbrev">TCS</title>
<idno type="ISSN">0304-3975</idno>
<imprint><publisher>ELSEVIER</publisher>
<date type="published" when="2001">2001</date>
<biblScope unit="volume">255</biblScope>
<biblScope unit="issue">1–2</biblScope>
<biblScope unit="page" from="687">687</biblScope>
<biblScope unit="page" to="696">696</biblScope>
</imprint>
<idno type="ISSN">0304-3975</idno>
</series>
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<seriesStmt><idno type="ISSN">0304-3975</idno>
</seriesStmt>
</fileDesc>
<profileDesc><textClass><keywords scheme="Teeft" xml:lang="en"><term>Attractivity problem</term>
<term>Blondel</term>
<term>Commutative diagram</term>
<term>Computational power</term>
<term>Continuous functions</term>
<term>Convergent</term>
<term>Decision algorithm</term>
<term>Discrete time</term>
<term>Dynamical</term>
<term>Dynamical state space</term>
<term>Dynamical system</term>
<term>Dynamical systems</term>
<term>Elsevier science</term>
<term>Global convergence</term>
<term>Hybrid</term>
<term>Hybrid systems</term>
<term>Immortal trajectory</term>
<term>Immortality problem</term>
<term>Initial point</term>
<term>Internal state</term>
<term>Internal states</term>
<term>Larger class</term>
<term>Linear systems</term>
<term>Main theorem</term>
<term>Metric space</term>
<term>Mortality problem</term>
<term>Nite number</term>
<term>Open problem</term>
<term>Simulation</term>
<term>Special state</term>
<term>State space</term>
<term>Study problems</term>
<term>Symmetrical argument</term>
<term>Theoretical computer science</term>
<term>Trajectory</term>
<term>Trajectory converges</term>
<term>Turing</term>
<term>Turing machine</term>
<term>Turing machines</term>
<term>Undecidability</term>
<term>Undecidable</term>
</keywords>
</textClass>
<langUsage><language ident="en">en</language>
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<front><div type="abstract" xml:lang="en">Abstract: In this paper we study problems such as: given a discrete time dynamical system of the form x(t+1)=f(x(t)) where f:Rn→Rn is a piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem that this Attractivity Problem is undecidable as soon as n⩾2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (do all trajectories go through 0?). We then show that Attractivity and Stability become decidable in dimension 1 for continuous functions.</div>
</front>
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Deciding stability and mortality of piecewise affine dynamical systems

Deciding stability and mortality of piecewise affine dynamical systems

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