The richness of phenomena observed in the load-deformation-temperature behaviour of shape memory alloys has provided a challenge for the physicist and mathematician as well as the engineer. The physicist is interested in understanding and explaining the phenomena in terms of the crystallographic structure of these metals. After understanding he will formulate models with the general aim of simulating the deformation as a response to a dynamic and thermal input. In the present case the basic notions needed for the modelization are those of statistical mechanics and thermodynamics. They lead to a system of non-linear ordinary differential equations, so that, given the load and temperature as functions of time, the deformation may be calculated as a function of time. The engineer is often interested in efficient actuators that allow him to control the shape of a body as a function of time. He wishes to calculate and apply the necessary load and temperature distributions that will realize that shape as quickly as possible and as efficiently as possible. The mathematician can help both the physicist and the engineer by providing the mathematical tools for the efficient solution of the model equations created by the physicist. Typically he will furnish the input functions which the engineer needs for the desired output. The mathematical tools required for this project are those of feedback control and optimal control.
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