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A thermodynamically consistent model of shape memory alloys in three dimensions is studied. The thermoelastic system, based on the strain tensor, its gradient and the absolute temperature, is a generalization of the well-known one-dimensional Falk model. The key assumptions concerning the form of constitutive relations are discussed. The detailed and selfcontained proof of the global-in-time existence and uniqueness of solutions is presented.
A thermodynamically consistent model of shape memory alloys in three dimensions is studied. The thermoelasticity system, based on the strain tensor, its gradient and the absolute temperature, generalizes the well-known one-dimensional Falk model. Under simplifying structural assumptions we prove global in time existence and uniqueness of the solution.
The convex optimal control problem for a system described by the parabolic equation is considered. The form of the right derivative of an optimal solution with respect to the parameter is derived. The applications to an air quality control problem are discussed. Numerical result are provided.
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