A State-Space Approach to a One-Dimensional Mathematical Model for the Dynamics of Phase Transitions in Pseudoelastic Materials

An abstract formulation of a general mathematical model for the dynamics of shape memory alloys is presented. Using a state-space approach, the model is written as a semilinear Cauchy problem in an appropriate Hilbert space. It is shown that the differential operator corresponding to the linear part of the system generates an exponentially stable analytic semigroup. For this semigroup, explicit decay rates are given in terms of the physical constants of the system and its continuity with respect to the model parameters is proved. A proof of local existence of solutions is given and open problems are discussed.