In this brief the authors establish a new frequency-sweeping framework to solve the complete stability problem for time-delay systems with commensurate delays. The text describes an analytic curve perspective which allows a deeper understanding of spectral properties focusing on the asymptotic behavior of the characteristic roots located on the imaginary axis as well as on properties invariant with respect to the delay parameters. This asymptotic behavior is shown to be related by another novel concept, the dual Puiseux series which helps make frequency-sweeping curves useful in the study of general time-delay systems. The comparison of Puiseux and dual Puiseux series leads to three important results:

• an explicit function of the number of unstable roots simplifying analysis and design of time-delay systems so that to some degree they may be dealt with as finite-dimensional systems;
• categorization of all time-delay systems into three types according to their ultimate stability properties; and
• a simple frequency-sweeping criterion allowing asymptotic behavior analysis of critical imaginary roots for all positive critical delays by observation. Academic researchers and graduate students interested in time-delay systems and practitioners working in a variety of fields - engineering, economics and the life sciences involving transfer of materials, energy or information which are inherently non-instantaneous, will find the results presented here useful in tackling some of the complicated problems posed by delays.

Autorentext
Silviu-Iulian Niculescu was born in Petrosani, Romania in 1968. He received the B.S. degree from the Polytechnical Institute of Bucharest, Romania, the M.Sc. and Ph.D. degrees, both in Automatic Control, from the Institut National Polytechnique de Grenoble, France and the Habilitation à Diriger des Recherches (HDR) in Automatic Control from Université de Technologie de Compiègne, in 1992, 1993, 1996 and 2003, respectively. From 1992 to 1997, he was with the Department of Automatic Control and Computers, University Politehnica Bucharest, Romania. From 1997 to 2006, he was with HEUDIASYC (Heuristics and diagnosis of complex systems) laboratory, Compiègne, France as a Researcher at CNRS (French National Center for Scientific Research). He also held a Post Doctoral position in the Department of Applied Mathematics, ENSTA, Paris, France, from 1996 to 1997. In September 2006, he joined L2S (Laboratory of Signals and Systems), Gif-sur-Yvette, where he is currently Research Director (Senior Researcher) at CNRS and head of the laboratory. He is the single author of two books: delay systems. Qualitative aspects on the stability and stabilization (Diderot: Paris, 1997; in French) and Delay effects on stability. A robust control approach (Springer: Heidelberg, LNCIS, vol. 269, 2001; in English), co-author of one book (with Wim Michiels) entitled Stability and stabilization of time-delay systems. An eigenvalue-based approach (SIAM: Philadelphia, USA, 2007) and co-editor of five volumes (SIAM, 1999; Pergamon Press, 2001; Springer 2004; Springer 2007; Springer 2009). He is author or co-author of more than 300 book chapters or scientific papers. He has been the IPC Chairman of the 3rd IFAC Workshop on Time-Delay Systems (Santa Fe, NM, USA, December 2001) and of the 8th IFAC Workshop on Time-Delay Systems (Sinaia, Romania, September 2009) and the main organizer or co-organizer of the 1st CNRS-NSF Workshop on Time-Delay Systems (Paris: La Défense,January 2003) and of the 23rd European Summer School in Automatic Control (Grenoble, September 2000) devoted to time-delay systems. He was the guest co-editor of five special issues in the area of delay systems (Journal of Mathematical Modelling of Natural Phenomena (MMNP) in 2009 Asian Journal of Control in 2005, IMA Journal of Mathematical Control and Information in 2002 and 2010 and Kybernetika in 2001). Dr. Niculescu has been scientific responsible of 14 international cooperation programs and projects (with Belgium, Mexico, Romania and Eastern European countries, South Korea, Turkey and United States). He is member of IFAC Technical Committee on Linear Systems (since 2002) and of the IPC of 33 International Conferences and he was an Associate Editor of the IEEE Transactions on Automatic Control (2003-2005). Since 2011, He is an Associate Editor of European Journal of Control and IMA Journal of Mathematical Control and Information. He is the responsible of the IFAC Research Group on Time-delay systems since its creation in October 2007. Dr. Niculescu was awarded the CNRS Bronze Medal for scientific research, the Best Paper Presentation Award at American Control Conference, Chicago, IL and the Ph.D. Thesis Award from INPG, Grenoble (France) in 2001, 2000, and 1996, respectively. His research interests include delay systems, robust control, operator theory and numerical methods in optimization and their applications to the design of engineering systems.

Inhalt
Introduction to the Stability of Time-Delay Systems.- Introduction to Analytic Curves.- Analytic Curve Perspective for Time-Delay Systems.- Computing Puiseux Series for a Critical Pair.- Invariance Property: A Unique Idea for Complete Stability Analysis.- Invariance Property for Critical Imaginary Roots (CIRs) with Index g = 1.- Invariance Property for Critical Imaginary Roots with Index n = 1.- Invariance Conjecture and Frequency-Sweeping Framework.- Complex Stability for Time-Delay Systems: A Unified Approach.- Extension to Neutral Time-Delay Systems.- Concluding Remarks and Further Perspectives.