Deciding stability and mortality of piecewise affine dynamical systems
Identifieur interne : 009206 ( Main/Exploration ); précédent : 009205; suivant : 009207Deciding 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.
Descripteurs français
- Pascal (Inist)
- Wicri :
- topic : Mortalité.
English descriptors
- KwdEn :
- Teeft :
- Attractivity problem, Blondel, Commutative diagram, Computational power, Continuous functions, Convergent, Decision algorithm, Discrete time, Dynamical, Dynamical state space, Dynamical system, Dynamical systems, Elsevier science, Global convergence, Hybrid, Hybrid systems, Immortal trajectory, Immortality problem, Initial point, Internal state, Internal states, Larger class, Linear systems, Main theorem, Metric space, Mortality problem, Nite number, Open problem, Simulation, Special state, State space, Study problems, Symmetrical argument, Theoretical computer science, Trajectory, Trajectory converges, Turing, Turing machine, Turing machines, Undecidability, Undecidable.
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
Affiliations:
- Belgique, France, États-Unis
- Auvergne-Rhône-Alpes, Californie, Grand Est, Lorraine (région), Massachusetts, Province du Brabant wallon, Rhône-Alpes, Région wallonne
- Louvain-la-Neuve, Lyon, Vandœuvre-lès-Nancy
- Université catholique de Louvain
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Le document en format XML
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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></TEI><affiliations><list><country><li>Belgique</li><li>France</li><li>États-Unis</li></country><region><li>Auvergne-Rhône-Alpes</li><li>Californie</li><li>Grand Est</li><li>Lorraine (région)</li><li>Massachusetts</li><li>Province du Brabant wallon</li><li>Rhône-Alpes</li><li>Région wallonne</li></region><settlement><li>Louvain-la-Neuve</li><li>Lyon</li><li>Vandœuvre-lès-Nancy</li></settlement><orgName><li>Université catholique de Louvain</li></orgName></list><tree><country name="Belgique"><region name="Région wallonne"><name sortKey="Blondel, Vincent D" sort="Blondel, Vincent D" uniqKey="Blondel V" first="Vincent D." last="Blondel">Vincent D. Blondel</name></region><name sortKey="Blondel, Vincent D" sort="Blondel, Vincent D" uniqKey="Blondel V" first="Vincent D." last="Blondel">Vincent D. Blondel</name></country><country name="France"><region name="Grand Est"><name sortKey="Bournez, Olivier" sort="Bournez, Olivier" uniqKey="Bournez O" first="Olivier" last="Bournez">Olivier Bournez</name></region><name sortKey="Bournez, Olivier" sort="Bournez, Olivier" uniqKey="Bournez O" first="Olivier" last="Bournez">Olivier Bournez</name><name sortKey="Koiran, Pascal" sort="Koiran, Pascal" uniqKey="Koiran P" first="Pascal" last="Koiran">Pascal Koiran</name><name sortKey="Koiran, Pascal" sort="Koiran, Pascal" uniqKey="Koiran P" first="Pascal" last="Koiran">Pascal Koiran</name></country><country name="États-Unis"><region name="Californie"><name sortKey="Papadimitriou, Christos H" sort="Papadimitriou, Christos H" uniqKey="Papadimitriou C" first="Christos H." last="Papadimitriou">Christos H. Papadimitriou</name></region><name sortKey="Papadimitriou, Christos H" sort="Papadimitriou, Christos H" uniqKey="Papadimitriou C" first="Christos H." last="Papadimitriou">Christos H. Papadimitriou</name><name sortKey="Tsitsiklis, John N" sort="Tsitsiklis, John N" uniqKey="Tsitsiklis J" first="John N." last="Tsitsiklis">John N. Tsitsiklis</name><name sortKey="Tsitsiklis, John N" sort="Tsitsiklis, John N" uniqKey="Tsitsiklis J" first="John N." last="Tsitsiklis">John N. Tsitsiklis</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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