WIAS Preprint No. 282, (1996)

Automatic control via thermostats of a hyperbolic Stefan problem with memory


  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Grasselli, Maurizio
  • Sprekels, Jürgen

2010 Mathematics Subject Classification

  • 35R35 35R70 45K05 93C20


  • Feedback control, Stefan problems, memory kernels, hyperbolic heat conduction




A hyperbolic Stefan problem based on the linearized Gurtin-Pipkin heat conduction law is considered. Temperature and free boundary are controlled by a thermostat acting on the boundary. This feedback control is based on temperature measurements performed by real thermal sensors located into the domain containing the two-phase system and/or at its boundary. Three different types of thermostats are analyzed: ideal switch, relay switch, and Preisach hysteresis operator. The resulting models lead to formulate integrodifferential hyperbolic Stefan problems with nonlinear and nonlocal boundary conditions. In all the cases, existence results are proved. Uniqueness is also shown, unless in the situation corresponding to the ideal switch.

Appeared in

  • Appl. Math. Optimiz., 39 (1999), pp. 229 - 255

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