A full characterization of the critical stability curves/surfaces of linear systems with multiple delays

The stability analysis of a linear system with the multiple delays as parameters in given intervals is not a new but hard topic in general, for which a key step is to find out all the critical stability curves/surfaces in the parameter space. In this paper, the critical stability condition is regarded as a complex equations depending nonlinearly on the delays, and it is solved in three parts: (1) The solvability of the nonlinear equation; (2) The representation of the solutions; 3) Numerical algorithms for finding the solutions. For the solvability, a necessary and sufficient condition in terms of a delay-independent inequality with clear geometrical meaning has been derived from the critical stability condition in the form of vector equation. For the representation, the critical delays in nested form are expressed explicitly in terms of a number of hypersurfaces, all the quantities have clear geometrical meaning. Based on the nested representation, two effective algorithms are proposed for finding the solutions, and illustrated with simple examples. The main results not only generalize the previous ones for systems with two delays and three delays of the nondegenerate cases, but also add new findings for the degenerated cases which have important impact on the stability of the time-delay systems.