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The Lambert \(W\) function and the spectrum of some multidimensional time-delay systems. (English) Zbl 1138.93026
Summary: We find an explicit expression for the eigenvalues of a retarded time-delay system with one delay, for the special case that the system matrices are simultaneously triangularizable, which includes the case where they commute. Using matrix function definitions we define a matrix version of the Lambert \(W\) function, from which we form the expression. We prove by counter-example that some expressions in other publications on Lambert \(W\) for time-delay systems do not always hold.

MSC:
93B60 Eigenvalue problems
93C35 Multivariable systems, multidimensional control systems
34K35 Control problems for functional-differential equations
Software:
DDE-BIFTOOL
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