Spectral analysis of a non-selfadjoint operator generated by an energy harvesting model and application to an exact controllability problem

Article type: Research Article

Authors: Shubov, Marianna A.

Affiliations: Department of Mathematics and Statistics, University of New Hampshire, 33 Academic Way, Durham, NH 03820, USA. E-mail: marianna.shubov@gmail.com

Abstract: The present paper is the second one in a series of three works dealing with a mathematical model for an electro-mechanical energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face. A pair of conductive electrodes, covering the top and bottom faces of the piezoceramic layer, are connected to a resistive load. The model is governed by a system of two differential equations. The first of them is the equation of the Euler–Bernoulli beam model subject to actions of an external damping and of an external force. The second equation represents the Kirchhoff’s law for the electric circuit. Both equations are coupled by means of the direct and converse piezoelectric effects. The system is represented as a single operator evolution equation in a Hilbert space (the state space). The dynamics generator of this system is a non-selfadjoint operator with a compact resolvent. In the first paper, the asymptotic formulas for the eigenvalues of the dynamics generator have been derived. In the present paper, the following results have been shown. 1) The set of the generalized eigenvectors of the dynamics generator is complete and minimal in the state space. 2) The set of normalized generalized eigenvectors forms a Riesz basis, which is quadratically close to an orthonormal basis (a Bari basis). 3) The exact controllability problem has been solved via the spectral decomposition method.

Keywords: Electro-mechanical energy harvester, non-selfadjoint matrix differential operator, generalized eigenvectors, Bari basis, exact and approximate controllability

DOI: 10.3233/ASY-171413

Journal: Asymptotic Analysis, vol. 102, no. 3-4, pp. 119-156, 2017