Port-Hamiltonian Structure of Interacting Particle Systems and Its Mean-Field Limit

Authors

Birgit Jacob, Claudia Totzeck

Abstract

We derive a minimal port-Hamiltonian formulation of a general class of interacting particle systems driven by alignment and potential-based force dynamics which include the Cucker-Smale model with potential interaction and the second order Kuramoto model. The port-Hamiltonian structure allows to characterize conserved quantities such as Casimir functions as well as the long-time behaviour using a LaSalle-type argument on the particle level. It is then shown that the port-Hamiltonian structure is preserved in the mean-field limit and an analogue of the LaSalle invariance principle is studied in the space of probability measures equipped with the 2-Wasserstein-metric. The results on the particle and mean-field limit yield a new perspective on uniform stability of general interacting particle systems. Moreover, as the minimal port-Hamiltonian formulation is closed we identify the ports of the subsystems which admit generalized mass-spring-damper structure modelling the binary interaction of two particles. Using the information of ports we discuss the coupling of difference species in a port-Hamiltonian preserving manner.

Citation

• Journal: Multiscale Modeling & Simulation
• Year: 2024
• Volume: 22
• Issue: 4
• Pages: 1247–1266
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/23m1547731

BibTeX

@article{Jacob_2024,
  title={{Port-Hamiltonian Structure of Interacting Particle Systems and Its Mean-Field Limit}},
  volume={22},
  ISSN={1540-3467},
  DOI={10.1137/23m1547731},
  number={4},
  journal={Multiscale Modeling \& Simulation},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Jacob, Birgit and Totzeck, Claudia},
  year={2024},
  pages={1247--1266}
}

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References

• Ahn, S. M., Choi, H., Ha, S.-Y. & Lee, H. On collision-avoiding initial configurations to Cucker-Smale type flocking models. Communications in Mathematical Sciences vol. 10 625–643 (2012) – 10.4310/cms.2012.v10.n2.a10
• Albi, G., Balagué, D., Carrillo, J. A. & von Brecht, J. Stability Analysis of Flock and Mill Rings for Second Order Models in Swarming. SIAM Journal on Applied Mathematics vol. 74 794–818 (2014) – 10.1137/13091779x
• Albi, G. & Pareschi, L. Modeling of self-organized systems interacting with a few individuals: From microscopic to macroscopic dynamics. Applied Mathematics Letters vol. 26 397–401 (2013) – 10.1016/j.aml.2012.10.011
• Ambrosio L.. Gradient Flows:in Metric Spaces and in the Space of Probability Measures (2005)
• Barbaro, A. B. T., Can͂izo, J. A., Carrillo, J. A. & Degond, P. Phase Transitions in a Kinetic Flocking Model of Cucker–Smale Type. Multiscale Modeling & Simulation vol. 14 1063–1088 (2016) – 10.1137/15m1043637
• Burger, M., Pinnau, R., Totzeck, C. & Tse, O. Mean-Field Optimal Control and Optimality Conditions in the Space of Probability Measures. SIAM Journal on Control and Optimization vol. 59 977–1006 (2021) – 10.1137/19m1249461
• Burger, M., Pinnau, R., Totzeck, C., Tse, O. & Roth, A. Instantaneous control of interacting particle systems in the mean-field limit. Journal of Computational Physics vol. 405 109181 (2020) – 10.1016/j.jcp.2019.109181
• Cao, F. et al. Asymptotic flocking for the three-zone model. Mathematical Biosciences and Engineering vol. 17 7692–7707 (2020) – 10.3934/mbe.2020391
• Carrillo, J. A., Fornasier, M., Toscani, G. & Vecil, F. Particle, kinetic, and hydrodynamic models of swarming. Modeling and Simulation in Science, Engineering and Technology 297–336 (2010) doi:10.1007/978-0-8176-4946-3_12 – 10.1007/978-0-8176-4946-3_12
• Carrillo, J. A., Hoffmann, F., Stuart, A. M. & Vaes, U. Consensus‐based sampling. Studies in Applied Mathematics vol. 148 1069–1140 (2022) – 10.1111/sapm.12470
• Carrillo, J. A., Choi, Y.-P., Mucha, P. B. & Peszek, J. Sharp conditions to avoid collisions in singular Cucker–Smale interactions. Nonlinear Analysis: Real World Applications vol. 37 317–328 (2017) – 10.1016/j.nonrwa.2017.02.017
• Carrillo, J. A., Choi, Y.-P. & Perez, S. P. A Review on Attractive–Repulsive Hydrodynamics for Consensus in Collective Behavior. Modeling and Simulation in Science, Engineering and Technology 259–298 (2017) doi:10.1007/978-3-319-49996-3_7 – 10.1007/978-3-319-49996-3_7
• Carrillo, J. A., Fornasier, M., Rosado, J. & Toscani, G. Asymptotic Flocking Dynamics for the Kinetic Cucker–Smale Model. SIAM Journal on Mathematical Analysis vol. 42 218–236 (2010) – 10.1137/090757290
• Carrillo, J. A., Fornasier, M., Toscani, G. & Vecil, F. Particle, kinetic, and hydrodynamic models of swarming. Modeling and Simulation in Science, Engineering and Technology 297–336 (2010) doi:10.1007/978-0-8176-4946-3_12 – 10.1007/978-0-8176-4946-3_12
• CAÑIZO, J. A., CARRILLO, J. A. & ROSADO, J. A WELL-POSEDNESS THEORY IN MEASURES FOR SOME KINETIC MODELS OF COLLECTIVE MOTION. Mathematical Models and Methods in Applied Sciences vol. 21 515–539 (2011) – 10.1142/s0218202511005131
• Cho, J., Ha, S.-Y., Huang, F., Jin, C. & Ko, D. Emergence of bi-cluster flocking for the Cucker–Smale model. Mathematical Models and Methods in Applied Sciences vol. 26 1191–1218 (2016) – 10.1142/s0218202516500287
• Choi, Y.-P. & Haskovec, J. Hydrodynamic Cucker–Smale Model with Normalized Communication Weights and Time Delay. SIAM Journal on Mathematical Analysis vol. 51 2660–2685 (2019) – 10.1137/17m1139151
• Cucker, F. & Dong, J.-G. A General Collision-Avoiding Flocking Framework. IEEE Transactions on Automatic Control vol. 56 1124–1129 (2011) – 10.1109/tac.2011.2107113
• Cucker, F. & Smale, S. On the mathematics of emergence. Japanese Journal of Mathematics vol. 2 197–227 (2007) – 10.1007/s11537-007-0647-x
• D’Orsogna, M. R., Chuang, Y. L., Bertozzi, A. L. & Chayes, L. S. Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse. Physical Review Letters vol. 96 (2006) – 10.1103/physrevlett.96.104302
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• Eberard, D., Maschke, B. M. & van der Schaft, A. J. An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics vol. 60 175–198 (2007) – 10.1016/s0034-4877(07)00024-9
• Erban, R., Haškovec, J. & Sun, Y. A Cucker–Smale Model with Noise and Delay. SIAM Journal on Applied Mathematics vol. 76 1535–1557 (2016) – 10.1137/15m1030467
• Golse, F. On the Dynamics of Large Particle Systems in the Mean Field Limit. Lecture Notes in Applied Mathematics and Mechanics 1–144 (2016) doi:10.1007/978-3-319-26883-5_1 – 10.1007/978-3-319-26883-5_1
• Ha, S.-Y., Kim, J., Park, J. & Zhang, X. Complete Cluster Predictability of the Cucker–Smale Flocking Model on the Real Line. Archive for Rational Mechanics and Analysis vol. 231 319–365 (2018) – 10.1007/s00205-018-1281-x
• Hinrichsen, D. & Pritchard, A. J. Mathematical Systems Theory I. Texts in Applied Mathematics (Springer Berlin Heidelberg, 2005). doi:10.1007/b137541 – 10.1007/b137541
• Matei I, Mavridis C, Baras JS, Zhenirovskyy M (2019) Inferring Particle Interaction Physical Models and Their Dynamical Properties. In: 2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, pp 4615–4621 – 10.1109/cdc40024.2019.9029524
• Motsch, S. & Tadmor, E. A New Model for Self-organized Dynamics and Its Flocking Behavior. Journal of Statistical Physics vol. 144 923–947 (2011) – 10.1007/s10955-011-0285-9
• Motsch, S. & Tadmor, E. Heterophilious Dynamics Enhances Consensus. SIAM Review vol. 56 577–621 (2014) – 10.1137/120901866
• Panaretos, V. M. & Zemel, Y. An Invitation to Statistics in Wasserstein Space. SpringerBriefs in Probability and Mathematical Statistics (Springer International Publishing, 2020). doi:10.1007/978-3-030-38438-8 – 10.1007/978-3-030-38438-8
• Park, J., Kim, H. J. & Ha, S.-Y. Cucker-Smale Flocking With Inter-Particle Bonding Forces. IEEE Transactions on Automatic Control vol. 55 2617–2623 (2010) – 10.1109/tac.2010.2061070
• Pignotti, C. & Trélat, E. Convergence to consensus of the general finite-dimensional Cucker–Smale model with time-varying delays. Communications in Mathematical Sciences vol. 16 2053–2076 (2018) – 10.4310/cms.2018.v16.n8.a1
• Reynolds, C. W. Flocks, herds and schools: A distributed behavioral model. ACM SIGGRAPH Computer Graphics vol. 21 25–34 (1987) – 10.1145/37402.37406
• Shvydkoy, R. Dynamics and Analysis of Alignment Models of Collective Behavior. Nečas Center Series (Springer International Publishing, 2021). doi:10.1007/978-3-030-68147-0 – 10.1007/978-3-030-68147-0
• Tanaka, H.-A., Lichtenberg, A. J. & Oishi, S. First Order Phase Transition Resulting from Finite Inertia in Coupled Oscillator Systems. Physical Review Letters vol. 78 2104–2107 (1997) – 10.1103/physrevlett.78.2104
• Teschl, G. Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics (2012) doi:10.1090/gsm/140 – 10.1090/gsm/140
• Totzeck, C. Trends in Consensus-Based Optimization. Modeling and Simulation in Science, Engineering and Technology 201–226 (2021) doi:10.1007/978-3-030-93302-9_6 – 10.1007/978-3-030-93302-9_6
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• van der Schaft A (2007) Port-Hamiltonian systems: an introductory survey. In: Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006. EMS Press, pp 1339–1365 – 10.4171/022-3/65
• Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. Novel Type of Phase Transition in a System of Self-Driven Particles. Physical Review Letters vol. 75 1226–1229 (1995) – 10.1103/physrevlett.75.1226