Piezoelectric material behavior is mathematically described by coupled hyperbolic-elliptic partial differential equations (PDEs) governing mechanical displacement and electrical potential.
This paper presents advancements in the theory of identifying material parameters in piezoelectric PDEs. We focus on modeling and analyzing the inverse problem assuming matrix-valued Sobolev-Bochner parameters to encompass a time and space-dependent setting and thus external physical influences. This is followed by results regarding the existence, uniqueness and improved regularity of solutions to the piezoelectric PDE. Based on these findings, well-definedness and regularity of the parameter-to-state map and Fréchet differentiability of the observation operator are proven. Finally, the inverse problem is formulated using a minimization approach, where weak lower semi-continuity of the objective functional, first-order optimality conditions and the derivation and analysis of the adjoint PDE are presented.