Appl Math Optim 39:229–255 (1999)
1999 Springer-Verlag New York Inc.
Automatic Control via Thermostats of a
Hyperbolic Stefan Problem with Memory
and J. Sprekels
Dipartimento di Matematica, Universit`a di Pavia,
Via Ferrata 1, I-27100 Pavia, Italy
Dipartimento di Matematica, Politecnico di Milano,
Via E. Bonardi 9, I-20133 Milano, Italy
Weierstraß-Institut f¨ur Angewandte Analysis und Stochastik (WIAS),
Mohrenstraße 39, D-10117 Berlin, Germany
Communicated by R. Triggiani
Abstract. A hyperbolic Stefan problem based on the linearized Gurtin–Pipkin
heat conduction law is considered. The temperature and free boundary are con-
trolled by a thermostat acting on the boundary. This feedback control is based on
temperature measurements performed by real thermal sensors located within the do-
main containing the two-phase system and/or at its boundary. Three different types
of thermostats are analyzed: simple switch, relay switch, and a Preisach hysteresis
operator. The resulting models lead to integrodifferential hyperbolic Stefan prob-
lems with nonlinear and nonlocal boundary conditions. Existence results are proved
in all the cases. Uniqueness is also shown, except in the situation corresponding to
the ideal switch.
Key Words. Hyperbolic Stefan problems, Heat conduction with memory, Ther-
mostat control, Hysteresis operators of Preisach type.
AMS Classiﬁcation. 35R35, 45K05, 80A22, 93B52.
Consider a two-phase system which occupies a bounded domain ⊂ R
(N ≥ 1) at
any time t ∈ [0, T] (T > 0). Letting Q
:= × (0, T ), we denote by ϑ: Q
→ R the
relative temperature (rescaled in order that ϑ = 0 is the critical temperature at which the