Exact versus discretized stability regions for a linear delay differential equation

摘要

The paper introduces a system of necessary and sufficient stability conditions for a four-term linear delay difference equation with complex coefficients. These conditions are derived explicitly with respect to the time lag and can be viewed as a direct discrete counterpart to the existing stability results for the underlying delay differential equation. As a main proof tool, the boundary locus technique combined with some special results of the polynomial theory is employed. Since the studied difference equation serves as a θ-method discretization of its continuous pattern, several problems of numerical stability are discussed as well.