Asymptotic and spectral analysis and control problems for mathematical model of piezoelectric energy harvester

Abstract

We present a set of our mathematical results on a model of piezoelectric energy harvester. The model is well known and is described in engineering literature. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler–Bernoulli model for the transverse vibrations of a beam subject to actions of an external (viscous/fluid) damping and of an external force. The second equation represents the Kirchhoff’s law for the electric circuit. Both equations are coupled
due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped–free type with an additional term that is proportional to the voltage on the resistive load and is produced by the converse piezoelectric effect. The circuit equation contains an extra term that
depends on the transverse displacement of the beam and is produced by the direct piezoelectric effect. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non–selfadjoint operator with compact resolvent. We present precise
statements and outline an approach to the proofs of the following results. 1) Asymptotic formulas  for the eigenvalues and the generalized eigenvectors of the dynamics generator. 2) Completeness, minimality, and Riesz basis property of the generalized eigenvectors. 3) Solutions by the spectral decomposition method of two control problems: zero controllability problem and a version of output tracking problem.