Well-defined series and parallel D-spectra for linear time-varying systems

An nth-order, scalar, linear time-varying (LTV) systems y(n)+Σ_k=1ⁿα_k(t)^(k-1)=0 can be conveniently represented as 𝒟_α{y}=0 using the scalar polynomial. Differential operator (SPDO) 𝒟_α=δⁿ+Σ_k=1ⁿα_k(t)δ^k-1, where δ=d/dt is the derivative operator. Based on a classical result of Floquet (1879) on the factorization of SPDO 𝒟_α=(δ-λ_n(t))...(δ-λ₁(t)), a unified eigenvalue theory has recently been developed. In that theory the collection {λ_k(t)}_k=1ⁿ are called a series D-spectrum (SD-spectrum) for 𝒟_α and an n-parameter family {ρ_k(t)=λ_1,k(t)}_k=1ⁿ are called a parallel D-spectrum (PD-spectrum) for 𝒟_α, where λ_1,k(t) are particular solutions for λ₁(t) satisfying some nonlinear independence constraints. Although more than a century old, the important Floquet factorization has apparently not been fully harnessed in LTV system theory and control, due mainly to the well-known fact that even for a 2nd-order time-invariant SPDO, the PD-eigenvalue ρ(t)=λ₁(t) satisfying the scalar Riccati equation λ˙₁+λ₁²+α₂λ₁+α₁λ₁=0 may suffer from finite-time singularities known as finite-escapes. In this paper, necessary and sufficient conditions for the existence of well-defined (free of finite-time singularities) SD- and PD-spectra for SPDOs with complex- and real-valued coefficients are presented. The new results will have significant impact on applications of the unified eigenvalue theory to the analysis and control of LTV systems, and its further development.