Optimal Control for Shape Memory Alloys of the One-Dimensional Frémond Model

Abstract

In this article, we consider optimal control problems for the one-dimensional Frémond model for shape memory alloys. This model is constructed in terms of basic functionals like free energy and pseudo-potential of dissipation. The state problem is expressed by a system of partial differential equations involving the balance equations for energy and momentum. We prove the existence of an optimal control that minimizes the cost functional for a nonlinear and nonsmooth state problem. Moreover, we show the necessary condition of the optimal pair by using optimal control problems for approximating systems.

Keywords:

• One-dimensional Frémond model
• optimal control problem
• shape memory alloys
• subdifferentials

AMS SUBJECT CLASSIFICATION:

• 49J20
• 35K55
• 35R35

Additional information

Funding

Pierluigi Colli gratefully acknowledges some support from the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) – Dept. of Mathematics “F. Casorati”, University of Pavia; the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica); the IMATI – C.N.R. Pavia, Italy. In addition, this work was supported by Grant-in-Aid for Scientific Research (C) No. 16K05224, 20K03672 (Ken Shirakawa), and 20K03665 (Noriaki Yamazaki), JSPS.