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The Scientific World Journal
Volume 2013, Article ID 274719, 6 pages
http://dx.doi.org/10.1155/2013/274719
Research Article

Controllability in Hybrid Kinetic Equations Modeling Nonequilibrium Multicellular Systems

Dipartimento di Scienze Matematiche, Politecnico, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 8 August 2013; Accepted 2 September 2013

Academic Editors: K. Ammari, M. M. Cavalcanti, and S. Sivasundaram

Copyright © 2013 Carlo Bianca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. A. Levin, “Complex adaptive systems: exploring the known, the unknown and the unknowable,” Bulletin of the American Mathematical Society, vol. 40, no. 1, pp. 3–19, 2003. View at Publisher · View at Google Scholar · View at Scopus
  2. D. J. Krause and G. D. Ruxton, Living in Groups, Oxford University Press, Oxford, UK, 2002.
  3. D. L. Abel and J. T. Trevors, “Self-organization vs. self-ordering events in life-origin models,” Physics of Life Reviews, vol. 3, no. 4, pp. 211–228, 2006. View at Publisher · View at Google Scholar · View at Scopus
  4. R. Eftimie, “Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review,” Journal of Mathematical Biology, vol. 65, no. 1, pp. 35–75, 2011. View at Publisher · View at Google Scholar · View at Scopus
  5. C. Bianca, “Kinetic theory for active particles modelling coupled to Gaussian thermostats,” Applied Mathematical Sciences, vol. 6, no. 13-16, pp. 651–660, 2012. View at Google Scholar · View at Scopus
  6. C. Bianca, “An existence and uniqueness theorem to the Cauchy problem for thermostatted-KTAP models,” International Journal of Mathematical Analysis, vol. 6, no. 17-20, pp. 813–824, 2012. View at Google Scholar
  7. C. Bianca, “Onset of nonlinearity in thermostatted active particles models for complex systems,” Nonlinear Analysis: Real World Applications, vol. 13, no. 6, pp. 2593–2608, 2012. View at Publisher · View at Google Scholar · View at Scopus
  8. C. Bianca, “Modeling complex systems by functional subsystems representation and thermostatted-KTAP methods,” Applied Mathematics & Information Sciences, vol. 6, pp. 495–499, 2012. View at Google Scholar
  9. C. Bianca, M. Ferrara, and L. Guerrini, “High-order moments conservation in thermostatted kinetic models,” Journal of Global Optimization, 2013. View at Publisher · View at Google Scholar
  10. C. Dogbe, “Nonlinear pedestrian-flow model: uniform well-posedness and global existence,” Applied Mathematics & Information Sciences, vol. 7, no. 1, pp. 29–40, 2013. View at Publisher · View at Google Scholar
  11. C. Dogbe, “On the modelling of crowd dynamics by generalized kinetic models,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 512–532, 2012. View at Publisher · View at Google Scholar · View at Scopus
  12. C. Bianca and C. Dogbe, “A mathematical model for crowd dynamics: multiscale analysis, fluctuations and random noise,” Nonlinear Studies, vol. 20, pp. 281–305, 2013. View at Google Scholar
  13. D. Helbing and P. Molnár, “Social force model for pedestrian dynamics,” Physical Review E, vol. 51, no. 5, pp. 4282–4286, 1995. View at Publisher · View at Google Scholar · View at Scopus
  14. V. Bagland, B. Wennberg, and Y. Wondmagegne, “Stationary states for the noncutoff Kac equation with a Gaussian thermostat,” Nonlinearity, vol. 20, no. 3, pp. 583–604, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. B. Wennberg and Y. Wondmagegne, “Stationary states for the Kac equation with a Gaussian thermostat,” Nonlinearity, vol. 17, no. 2, pp. 633–648, 2004. View at Publisher · View at Google Scholar · View at Scopus
  16. B. Wennberg and Y. Wondmagegne, “The Kac equation with a thermostatted force field,” Journal of Statistical Physics, vol. 124, no. 2-4, pp. 859–880, 2006. View at Publisher · View at Google Scholar · View at Scopus
  17. P. Degond and B. Wennberg, “Mass and energy balance laws derived from high-field limits of thermostated Boltzmann equations,” Communications in Mathematical Sciences, vol. 5, no. 2, pp. 355–382, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. C. Bianca, “Thermostatted kinetic equations as models for complex systems in physics and life sciences,” Physics of Life Reviews, vol. 9, no. 4, pp. 359–399, 2012. View at Publisher · View at Google Scholar
  19. D. J. Evans, W. G. Hoover, B. H. Failor, B. Moran, and A. J. C. Ladd, “Nonequilibrium molecular dynamics via Gauss's principle of least constraint,” Physical Review A, vol. 28, no. 2, pp. 1016–1021, 1983. View at Publisher · View at Google Scholar · View at Scopus
  20. D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Fluids, Academic Press, New York, NY, USA, 1990.
  21. O. G. Jepps and L. Rondoni, “Deterministic thermostats, theories of nonequilibrium systems and parallels with the ergodic condition,” Journal of Physics A, vol. 43, no. 13, Article ID 133001, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. G. P. Morriss and C. P. Dettmann, “Thermostats: analysis and application,” Chaos, vol. 8, no. 2, pp. 321–336, 1998. View at Google Scholar · View at Scopus
  23. D. Ruelle, “Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics,” Journal of Statistical Physics, vol. 95, no. 1-2, pp. 393–468, 1999. View at Publisher · View at Google Scholar · View at Scopus
  24. S. Sivasundaram and J. Uvah, “Controllability of impulsive hybrid integro-differential systems,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 4, pp. 1003–1009, 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. K. F. Gauss, “On a New Fundamental Law of Mechanics,” Journal für die Reine und Angewandte Mathematik, vol. 4, pp. 232–235, 1829. View at Google Scholar
  26. M. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Belknap Press, 2006.
  27. C. Bianca and M. Pennisi, “The triplex vaccine effects in mammary carcinoma: a nonlinear model in tune with SimTriplex,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1913–1940, 2012. View at Publisher · View at Google Scholar · View at Scopus
  28. M. Pennisi, F. Pappalardo, and S. Motta, “Agent based modeling of lung metastasis-immune system competition,” Lecture Notes in Computer Science, vol. 5666, pp. 1–3, 2009. View at Publisher · View at Google Scholar · View at Scopus
  29. R. M. May, “Uses and abuses of mathematics in biology,” Science, vol. 303, no. 5659, pp. 790–793, 2004. View at Publisher · View at Google Scholar · View at Scopus
  30. A. Bellouquid and C. Bianca, “Modelling aggregation-fragmentation phenomena from kinetic to macroscopic scales,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 802–813, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. C. Bianca, “Existence of stationary solutions in kinetic models with Gaussian thermostats,” Mathematical Methods in the Applied Sciences, vol. 36, no. 13, pp. 1768–1775, 2013. View at Publisher · View at Google Scholar
  32. M. A. Ragusa, “Commutators of fractional integral operators on Vanishing-Morrey spaces,” Journal of Global Optimization, vol. 40, no. 1-3, pp. 361–368, 2008. View at Publisher · View at Google Scholar · View at Scopus