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Dependence on the Initial Data for the Continuous Thermostatted Framework

Author

Listed:
  • Bruno Carbonaro

    (Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy)

  • Marco Menale

    (Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy
    Laboratoire Quartz EA 7393, École Supérieure d’Ingénieurs en Génie Électrique, Productique et Management Industriel, 95092 Cergy Pontoise Cedex, France)

Abstract

The paper deals with the problem of continuous dependence on initial data of solutions to the equation describing the evolution of a complex system in the presence of an external force acting on the system and of a thermostat, simply identified with the condition that the second order moment of the activity variable (see Section 1) is a constant. We are able to prove that these solutions are stable with respect to the initial conditions in the Hadamard’s sense. In this connection, two remarks spontaneously arise and must be carefully considered: first, one could complain the lack of information about the “distance” between solutions at any time t ∈ [ 0 , + ∞ ) ; next, one cannot expect any more complete information without taking into account the possible distribution of the transition probabiliy densities and the interaction rates (see Section 1 again). This work must be viewed as a first step of a research which will require many more steps to give a sufficiently complete picture of the relations between solutions (see Section 5).

Suggested Citation

  • Bruno Carbonaro & Marco Menale, 2019. "Dependence on the Initial Data for the Continuous Thermostatted Framework," Mathematics, MDPI, vol. 7(7), pages 1-11, July.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:7:p:602-:d:246268
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    References listed on IDEAS

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    1. Kacperski, Krzysztof & Hoł yst, Janusz A., 1999. "Opinion formation model with strong leader and external impact: a mean field approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(2), pages 511-526.
    2. Carlo Bianca & Caterina Mogno, 2018. "Modelling pedestrian dynamics into a metro station by thermostatted kinetic theory methods," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 24(2), pages 207-235, March.
    3. Masurel, Léon & Bianca, Carlo & Lemarchand, Annie, 2018. "On the learning control effects in the cancer-immune system competition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 462-475.
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    Cited by:

    1. Carlo Bianca, 2022. "On the Modeling of Energy-Multisource Networks by the Thermostatted Kinetic Theory Approach: A Review with Research Perspectives," Energies, MDPI, vol. 15(21), pages 1-22, October.
    2. Carlo Bianca & Marco Menale, 2022. "On the Existence of Self-Similar Solutions in the Thermostatted Kinetic Theory with Unbounded Activity Domain," Mathematics, MDPI, vol. 10(9), pages 1-14, April.

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