Spectral characteristics and eigenvalues computation of the harmonic state operators in continuous-time periodic systems

Abstract

Spectral characteristics of what we call the harmonic state operators in finite-dimensional linear continuous-time periodic (FDLCP) systems are examined by means of the Fredholm theory for the first time. It is shown that the harmonic state operator is a closed, densely defined Fredholm operator on the Hilbert space l 2, and its spectrum contains only eigenvalues of finite type in any bounded neighborhood of the origin of the complex plane. These spectral characteristics lead us to an asymptotic computation algorithm for the eigenvalues of the harmonic state operator via truncations on its Fredholm regularization. Furthermore, the truncation approach also gives us a necessary and sufficient stability criterion in the FDLCP setting, which only involves the Fourier coefficients of the state matrix. Asymptotic stability of the lossy Mathieu differential equation is investigated to illustrate the results.