Complete characterization of the asymptotic stability region for single-delay second-order linear systems

Abstract

Determining the exact set of parameters for which a linear time-delay system is asymptotically stable remains a central problem in the analysis of retarded functional differential equations, and complete, closed-form answers are scarce even in low dimensions. We consider a second-order retarded functional differential equation with four real parameters and a single commensurate delay. The stability region is derived constructively by combining Pontryagin’s classical theorem on quasi-polynomials with a novel geometric characterization based on the supremum envelope of a family of parametric curves; an accompanying algorithm computes the resulting boundary explicitly. We obtain necessary and sufficient conditions identifying the exact region of the parameter space for which the trivial solution is asymptotically stable. For particular sub-classes of systems, the characterization reduces to very simple closed-form stability criteria, illustrated by examples including a delay-independent stabilization condition for an internal combustion engine model. Since the boundary is available in explicit form, robustness indices follow naturally as distances to it, and the algorithm supports standard visualization and control-design workflows.