Citation: Mattia Bongini, Massimo Fornasier. Sparse stabilization of dynamical systems driven by attraction and avoidance forces[J]. Networks and Heterogeneous Media, 2014, 9(1): 1-31. doi: 10.3934/nhm.2014.9.1
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Cluster algebras were invented by Fomin and Zelevinsky in a series of papers [9,2,10,11]. A cluster algebra is a
We first recall the definition of cluster automorphisms, which were introduced by Assem, Schiffler and Shamchenko in [1].
Definition 1.1 ([1]). Let
Cluster automorphisms and their related groups were studied by many authors, and one can refer to [6,7,8,14,13,4,5,16] for details.
The following very insightful conjecture on cluster automorphisms is by Chang and Schiffler, which suggests that we can weaken the conditions in Definition 1.1. In particular, it suggests that the second condition in Definition 1.1 can be obtained from the first one and the assumption that
Conjecture 1. [5,Conjecture 1] Let
The following is our main result, which affirms the Conjecture 1.
Theorem 3.6 Let
In this section, we recall basic concepts and important properties of cluster algebras. In this paper, we focus on cluster algebras without coefficients (that is, with trivial coefficients). For a positive integer
Recall that
Fix an ambient field
$ x'_k = \frac{\prod\limits_{i = 1}^n x_i^{[b_{ik}]_+} + \prod\limits_{i = 1}^n x_i^{[-b_{ik}]_+}}{x_k} $ |
and
$ b'_{ij} = {−bij,if i=k or j=k;bij+sgn(bik)[bikbkj]+,otherwise. $
|
where
It can be seen that
Let
Lemma 2.1 ([2]). Let
(1)
(2)
(3)
Definition 2.2 ([9,11]).
$ {\bf{x}}_t = (x_{1;t}, x_{2;t} ,\ldots,x_{n;t}),\,\,\,\,\,\,B_t = (b_{ij}^t). $ |
The cluster algebra
Theorem 2.3 (Laurent phenomenon and positivity [11,15,12]). Let
$ \mathbb{Z}_{\geq 0}[x_{1;t_0}^{\pm 1},x_{2;t_0}^{\pm1},\dots,x_{n;t_0}^{\pm1}]. $ |
In this section, we will give our main result, which affirms the Conjecture 1.
Lemma 3.1. Let
Proof.. Since
$ B^\prime = aE(B^\prime)F = a^2E^2(B^\prime)F^2 = \cdots = a^sE^s(B^\prime)F^s, $ |
where
Assume by contradiction that
A square matrix
Lemma 3.2. Let
Proof. If there exists
Let
$ BD = B^\prime AD = (B^\prime D)A. $ |
By the definition of mutation, we know that
Lemma 3.3. Let
Proof. By the definition of mutation, we know that
Lemma 3.4. Let
Proof. After permutating the rows and columns of
$ B = diag(B_1,B_2,\cdots,B_s), $ |
where
Without loss of generality, we assume that
Let
$ x_kx_k^\prime = \prod\limits_{i = 1}^nx_i^{[b_{ik}]_+}+\prod\limits_{i = 1}^nx_i^{[-b_{ik}]_+},\;\;\;\text{and}\;\;\; z_kz_k^\prime = \prod\limits_{i = 1}^nz_i^{[b_{ik}^\prime]_+}+\prod\limits_{i = 1}^nz_i^{[-b_{ik}^\prime]_+}. $ |
Thus
$ f(x′k)=f(n∏i=1x[bik]+i+n∏i=1x[−bik]+ixk)=n∏i=1z[bik]+i+n∏i=1z[−bik]+izk=n∏i=1z[bik]+i+n∏i=1z[−bik]+in∏i=1z[b′ik]+i+n∏i=1z[−b′ik]+iz′k. $
|
Note that the above expression is the expansion of
$ f(x_k^\prime) \in f(\mathcal A) = \mathcal{A} \subset \mathbb Z[z_1^{\pm 1}, \dots , (z'_k)^{\pm 1}, \dots , z_n^{\pm 1}], $ |
we can get
$ \frac{\prod\limits_{i = 1}^nz_i^{[b_{ik}]_+}+\prod\limits_{i = 1}^nz_i^{[-b_{ik}]_+}} {\prod\limits_{i = 1}^nz_i^{[b_{ik}^\prime]_+}+\prod\limits_{i = 1}^nz_i^{[-b_{ik}^\prime]_+}}\in\mathbb Z[z_1^{\pm1},\cdots,z_{k-1}^{\pm1},z_{k+1}^{\pm1},\cdots,z_n^{\pm1}]. $ |
Since both
$ \frac{\prod\limits_{i = 1}^nz_i^{[b_{ik}]_+}+\prod\limits_{i = 1}^nz_i^{[-b_{ik}]_+}} {\prod\limits_{i = 1}^nz_i^{[b_{ik}^\prime]_+}+\prod\limits_{i = 1}^nz_i^{[-b_{ik}^\prime]_+}}\in\mathbb Z[z_1,\cdots,z_{k-1},z_{k+1},\cdots,z_n]. $ |
So for each
$ B = diag(B_1,B_2,\cdots,B_s), $ |
where
$ (b_{1k}, b_{2k}, \dots , b_{nk})^{\rm T} = a_k(b'_{1k}, b'_{2k}, \dots , b'_{nk})^{\rm T} = \pm (b'_{1k}, b'_{2k}, \dots , b'_{nk})^{\rm T}. $ |
Hence,
$ \frac{\prod\limits_{i = 1}^nz_i^{[b_{ik}]_+}+\prod\limits_{i = 1}^nz_i^{[-b_{ik}]_+}} {\prod\limits_{i = 1}^nz_i^{[b_{ik}^\prime]_+}+\prod\limits_{i = 1}^nz_i^{[-b_{ik}^\prime]_+}} = 1. $ |
Thus we get
$ f(x′k)=n∏i=1z[bik]+i+n∏i=1z[−bik]+in∏i=1z[b′ik]+i+n∏i=1z[−b′ik]+iz′k=z′k. $
|
So
Lemma 3.5. Let
(i)
(ii)
Proof. (ⅰ) Since
Since
Hence,
(ⅱ) follows from (ⅰ) and the definition of cluster automorphisms.
Theorem 3.6. Let
Proof. "Only if part": It follows from the definition of cluster automorphism.
"If part": It follows from Lemma 3.4 and Lemma 3.5.
This project is supported by the National Natural Science Foundation of China (No.11671350 and No.11571173) and the Zhejiang Provincial Natural Science Foundation of China (No.LY19A010023).
[1] | L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford University Press, New York, 2000. |
[2] |
J.-P. Aubin and A. Cellina, Differential Inclusions, Set-valued maps and viability theory, Grundlehren der Mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4
![]() |
[3] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Math. Control Relat. Fields, 3 (2013), 447-466. Available from: http://www-m15.ma.tum.de/foswiki/pub/M15/Allgemeines/PublicationsEN/flocking_V9.pdf. doi: 10.3934/mcrf.2013.3.447
![]() |
[4] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, CISM International Centre for Mechanical Sciences, 553, Springer, 2014, 1-46. doi: 10.1007/978-3-7091-1785-9_1
![]() |
[5] |
J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363
![]() |
[6] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2010, 297-336. doi: 10.1007/978-0-8176-4946-3_12
![]() |
[7] |
Y. Chuang, M. D'Orsogna, D. Marthaler, A. Bertozzi and L. Chayes, State transition and the continuum limit for the 2D interacting, self-propelled particle system, Physica D, 232 (2007), 33-47. doi: 10.1016/j.physd.2007.05.007
![]() |
[8] |
F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129. doi: 10.1109/TAC.2011.2107113
![]() |
[9] |
F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete and Continuous Dynamical Systems, 34 (2014), 1009-1020. doi: 10.3934/dcds.2014.34.1009
![]() |
[10] |
F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842
![]() |
[11] |
F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x
![]() |
[12] |
M. D'Orsogna, Y. Chuang, A. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.104302
![]() |
[13] | A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. |
[14] | M. Fornasier and F. Solombrino, Mean-field optimal control, preprint, arXiv:1306.5913, (2013). |
[15] |
S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113
![]() |
[16] | J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. |
[17] | M. Huang, P. Caines and R. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and nash equilibrium solutions, in Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii, USA, December, 2003, 98-103. |
[18] |
J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math. (3), 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8
![]() |
[19] | SIAM Rev., to appear. |
[20] |
M. Nuorian, P. Caines and R. Malhamé, Synthesis of Cucker-Smale type flocking via mean field stochastic control theory: Nash equilibria, in Proceedings of the 48th Allerton Conf. on Comm., Cont. and Comp., Monticello, Illinois, 2010, 814-819. doi: 10.1109/ALLERTON.2010.5706992
![]() |
[21] | M. Nuorian, P. Caines and R. Malhamé, Mean field analysis of controlled Cucker-Smale type flocking: Linear analysis and perturbation equations, in Proceedings of 18th IFAC World Congress Milano (Italy) August 28-September 2, 2011, 4471-4476. |
[22] |
A. Rahmani, M. Ji, M. Mesbahi and M. Egerstedt, Controllability of multi-agent systems from a graph-theoretic perspective, SIAM J. Control and Optimization, 48 (2009), 162-186. doi: 10.1137/060674909
![]() |
[23] |
H. G. Tanner, On the controllability of nearest neighbor interconnections, in Proceedings of the 43rd IEEE Conference on Decision and Control, IEEE Press, Piscataway, NJ, 2004, 2467-2472. doi: 10.1109/CDC.2004.1428782
![]() |
[24] |
T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004
![]() |
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