Document Zbl 1405.93014

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Appl. Math.</a> <a href="/?q=in%3A426650" title="Articles in this Issue">79, No. 1, 95-110 (2019)</a>. </div> <div class="abstract">Summary: We prove unconditional and conditional flocking results for a Cucker-Smale flocking model enhanced with switching topologies. These topologies can be arbitrary; in particular, they are not necessarily symmetric. Our proofs hold for both discrete and continuous time. In both cases, the critical exponent discriminating between unconditional and conditional flocking is shown to be at most $\frac{1}{2(k-1)}$, where $k$ is the number of agents in the population.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7005245">Cited in <strong>16</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93A14" title="MSC2020">93A14</a> </td> <td class="space"> Decentralized systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C15" title="MSC2020">93C15</a> </td> <td class="space"> Control/observation systems governed by ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C30" title="MSC2020">93C30</a> </td> <td class="space"> Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C20" title="MSC2020">05C20</a> </td> <td class="space"> Directed graphs (digraphs), tournaments </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ACucker-Smale+model">Cucker-Smale model</a>; <a href="/?q=ut%3Aflocking">flocking</a>; <a href="/?q=ut%3Aswitching+topology">switching topology</a></div> <div class="software"> <h3>Software:</h3><a href="/software/38863">Boids</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1405.93014" data-ciurl="/ci/07005245" data-biburl="/bibtex/07005245.bib" data-amsurl="/amsrefs/07005245.bib" data-xmlurl="/xml/07005245.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07005245.pdf" title="Zbl 1405.93014 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1137/18M116976X" aria-label="DOI for “On flocks under switching directed interaction topologies”" title="10.1137/18M116976X">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, {\it Interaction ruling animal collective behaviour depends on topological rather than metric distance: Evidence from a field study}, Proc. Natl. Acad. Sci., 105 (2008), pp. 1232-1237.</td> </tr><tr> <td>[2]</td> <td class="space">A. Blanchet and P. Degond, {\it Kinetic models for topological nearest-neighbor interactions}, J. Statist. Phys., 169 (2017), pp. 929-950. · <a href="/1380.82032" class="nowrap">Zbl 1380.82032</a></td> </tr><tr> <td>[3]</td> <td class="space">M. Cao, A. S. Morse, and B. D. O. Anderson, {\it Reaching a consensus in a dynamically changing environment: A graphical approach}, SIAM J. Control Optim., 47 (2008), pp. 575-600. · <a href="/1157.93514" class="nowrap">Zbl 1157.93514</a></td> </tr><tr> <td>[4]</td> <td class="space">D. Chen, X. Liu, B. Xu, and H.-T. Zhang, {\it Intermittence and connectivity of interactions in pigeon flock flights}, Sci. Rep., 7 (2017), 10452.</td> </tr><tr> <td>[5]</td> <td class="space">I. D. Couzin, J. Krause, N. R. Franks, and S. Levin, {\it Effective leadership and decision making in animal groups on the move}, Nature, 433 (2005), pp. 513-516.</td> </tr><tr> <td>[6]</td> <td class="space">F. Cucker and J.-G. Dong, {\it On the critical exponent for flocks under hierarchical leadership}, Math. Models Methods Appl. Sci., 19 (2009), pp. 1391-1404. · <a href="/1202.92089" class="nowrap">Zbl 1202.92089</a></td> </tr><tr> <td>[7]</td> <td class="space">F. Cucker and J.-G. Dong, {\it On flocks influenced by closest neighbors}. Math. Models Methods Appl. Sci., 26 (2016), pp. 2685-2708. · <a href="/1354.93069" class="nowrap">Zbl 1354.93069</a></td> </tr><tr> <td>[8]</td> <td class="space">F. Cucker and S. Smale, {\it Emergent behavior in flocks}, IEEE Trans. Automat. Control, 52 (2007), pp. 852-862. · <a href="/1366.91116" class="nowrap">Zbl 1366.91116</a></td> </tr><tr> <td>[9]</td> <td class="space">F. Cucker and S. Smale, {\it On the mathematics of emergence}, Japan. J. Math., 2 (2007), pp. 197-227. · <a href="/1166.92323" class="nowrap">Zbl 1166.92323</a></td> </tr><tr> <td>[10]</td> <td class="space">F. Dalmao and E. Mordecki, {\it Hierarchical Cucker-Smale model subject to random failure}, IEEE Trans. Automat. Control, 57 (2012), pp. 1124-1129. · <a href="/1369.93005" class="nowrap">Zbl 1369.93005</a></td> </tr><tr> <td>[11]</td> <td class="space">J.-G. Dong and L. Qiu, {\it Flocking of the Cucker-Smale model on general digraphs}, IEEE Trans. Automat. Control, 62 (2017), pp. 5234-5239. · <a href="/1390.93031" class="nowrap">Zbl 1390.93031</a></td> </tr><tr> <td>[12]</td> <td class="space">D. Floreano and R. J. Wood, {\it Science, technology and the future of small autonomous drones}, Nature, 521 (2015), pp. 460-466.</td> </tr><tr> <td>[13]</td> <td class="space">A. Jadbabaie, J. Lin, and A. Morse, {\it Coordination of groups of mobile autonomous agents using nearest neighbor rules}, IEEE Trans. Automat. Control, 48 (2003), pp. 988-1001. · <a href="/1364.93514" class="nowrap">Zbl 1364.93514</a></td> </tr><tr> <td>[14]</td> <td class="space">Z. Li, S.-Y. Ha, and X. Xue, {\it Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership}, Math. Models Methods Appl. Sci., 24 (2014), pp. 1389-1419. · <a href="/1326.92075" class="nowrap">Zbl 1326.92075</a></td> </tr><tr> <td>[15]</td> <td class="space">Z. Li and X. Xue, {\it Cucker-Smale flocking under rooted leadership with fixed and switching topologies}, SIAM J. Appl. Math., 70 (2010), pp. 3156-3174. · <a href="/1209.92064" class="nowrap">Zbl 1209.92064</a></td> </tr><tr> <td>[16]</td> <td class="space">L. Moreau, {\it Stability of continuous-time distributed consensus algorithms}, in Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, pp. 3998-4003.</td> </tr><tr> <td>[17]</td> <td class="space">L. Moreau, {\it Stability of multiagent systems with time-dependent communication links}, IEEE Trans. Automat. Control, 50 (2005), pp. 169-182. · <a href="/1365.93268" class="nowrap">Zbl 1365.93268</a></td> </tr><tr> <td>[18]</td> <td class="space">S. Motsch and E. Tadmor, {\it A new model for self-organized dynamics and its flocking behavior}, J. Statist. Phys., 144 (2011), pp. 923-947. · <a href="/1230.82037" class="nowrap">Zbl 1230.82037</a></td> </tr><tr> <td>[19]</td> <td class="space">M. Nagy, Z. Ákos, D. Biro, and T. Vicsek, {\it Hierarchical group dynamics in pigeon flocks}, Nature, 464 (2010), pp. 890-893.</td> </tr><tr> <td>[20]</td> <td class="space">L. Perea, P. Elosegui, and G. Gómez, {\it Extension of the Cucker-Smale control law to space flight formations}, J. Guidance Contr. Dyn., 32 (2009), pp. 526-536.</td> </tr><tr> <td>[21]</td> <td class="space">B. Pettit, Z. Ákos, T. Vicsek, and D. Biro, {\it Speed determines leadership and leadership determines learning during pigeon flocking}, Current Biology, 25 (2015), pp. 3132-3137.</td> </tr><tr> <td>[22]</td> <td class="space">W. Ren and R.W. Beard, {\it Consensus seeking in multiagent systems under dynamically changing interaction topologies}, IEEE Trans. Automat. Control, 50 (2005), pp. 655-661. · <a href="/1365.93302" class="nowrap">Zbl 1365.93302</a></td> </tr><tr> <td>[23]</td> <td class="space">C. W. Reynolds, {\it Flocks, herds, and schools: A distributed behavioral model}, Computer Graphics, 21 (1987), pp. 25-34.</td> </tr><tr> <td>[24]</td> <td class="space">N. Rouche, P. Habets, and M. Laloy, {\it Stability Theory by Liapunov’s Direct Method}, Springer-Verlag, New York, 1977. · <a href="/0364.34022" class="nowrap">Zbl 0364.34022</a></td> </tr><tr> <td>[25]</td> <td class="space">J. Shen, {\it Cucker-Smale flocking under hierarchical leadership}, SIAM J. Appl. Math., 68 (2007), pp. 694-719. · <a href="/1311.92196" class="nowrap">Zbl 1311.92196</a></td> </tr><tr> <td>[26]</td> <td class="space">E. D. Sontag, {\it Mathematical Control Theory: Deterministic Finite Dimensional Systems}, 2nd ed., Springer-Verlag, New York, 1998. · <a href="/0945.93001" class="nowrap">Zbl 0945.93001</a></td> </tr><tr> <td>[27]</td> <td class="space">T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, {\it Novel type of phase transition in a system of self-driven particles}, Phys. Rev. Lett., 75 (1995), pp. 1226-1229.</td> </tr><tr> <td>[28]</td> <td class="space">T. Vicsek and A. Zefeiris, {\it Collective motion}, Phys. Rep., 517 (2012), pp. 71-140.</td> </tr><tr> <td>[29]</td> <td class="space">C. W. Wu, {\it Synchronization and convergence of linear dynamics in random directed networks}, IEEE Trans. Automat. Control, 51 (2006), pp. 1207-1210. · <a href="/1366.93537" class="nowrap">Zbl 1366.93537</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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