Invariants of Linear Time-Varying Dynamical Systems under D-Similarity Transformations

In a series of recent papers [11], [12], the new notions of ED-eigenvalues, D-spectra, D-characteristic equation, and D-similarity transformations were introduced for (time-varying) Vector Linear Dynamical Systems (VLDS) of the form ¿= A(t)x and Scalar LDS (SLDS) of the form y⁽ⁿ⁾ + ¿ⁿk=1 ¿k(t)y^(k-1)=0. The present paper establishes several fundamental features of D-similarity classes of (time-varying) VLDS or SLDS, and thereby answers some important questions about uniqueness, and invariance under D-similarity transformation, of ED-eigenvalues, D-spectra, etc.. The results obtained in this paper are useful in deriving analytical solutions and stability criteria for time-varying VLDS and SLDS. Several worked examples are included here to demonstrate such applications.