Finite-Dimensional Autonomous ODE Realizations of a Delay Differential Equation: The Role of Initialization Regularity and Observability (HDG_New1)

This preprint investigates the conditions under which the future trajectories of a scalar linear delay differential equation (DDE), initialized by a finite-dimensional class of polynomial histories, can be reproduced exactly by a finite-dimensional autonomous ordinary differential equation (ODE). Focusing on the DDE x˙(t)=−x(t)+12x(t−1)x˙(t)=−x(t)+21x(t−1) with the history class Φsub=span⁡{1,θ,θ2}Φsub=span{1,θ,θ2}, we establish rigorous boundaries between realizability and non-realizability. We prove that under differentiable initialization, exact reproduction is impossible. This is demonstrated by comparing the variational ODE output, which has a rational Laplace transform with finitely many poles, to the DDE response, which possesses infinitely many distinct, uncancelled simple poles (characterized by the branches of the Lambert W function). Conversely, if the differentiability of the initialization map is dropped entirely, we construct an exact five-dimensional realization using a discontinuous initialization map and a switching output function. Furthermore, we show that restoring continuity of the initialization map, combined with full observability of the linearized system, allows us to recover the differentiability of the initialization map from the exact reproduction condition, thereby reinstating the impossibility result. Finally, we analyze the case of partial observability, reducing the nonlinear compensation problem to a full-time compatibility condition that remains open in general, and we prove a model-specific non-realizability result for a minimal quadratic system with one hidden coordinate. These results precisely delineate the regularity and observability assumptions that are essential for the non-realizability of delay dynamics by finite-dimensional autonomous ODEs.