Parallel D-eigenvalues and parallel D-eigenvectors for linear time-varying systems

Singular behavior of PD-eigenvalues of an nth-order scalar polynomial differential operator (SPDO) 𝒟_α=δ ⁿ+Σ_k=1ⁿα_k (t)δ^k-1, where δ=d/dt, is investigated. Main results of this paper include: (i) definition and properties of PD-eigenvectors associated with PD-eigenvalues, (ii) definitions and properties of generalized PD-eigenvectors and generalized PD-eigenvalues for singular PD-eigenvalues; (iii) application of (i) and (ii) in stability analysis of linear time-varying (LTV) systems 𝒟_α{y}=0, and (iv) application (i) and (ii) in the realization of PD-characteristic equation. The new results will have a significant impact on applications of the unified eigenvalue theory to the analysis and control of LTV control systems, and its further development