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Multiplicity-induced-dominancy for delay-differential equations of retarded type. (English) Zbl 1471.34140

Consider a delay equation of the form \[ y^{(n)}(t)+\sum_{k=0}^{n-1}a_ky^{(k)}(t)+ \sum_{k=0}^{n-1}\alpha_k y^{(k)}(t-\tau)=0 \] with real coefficients and \(\tau>0\). The related characteristic function is defined as \[ \Delta(s)=s^n+\sum_{k=0}^{n-1}a_k+ e^{-s\tau}\sum_{k=0}^{n-1}\alpha_ks^k. \] Let \(s_0\in\mathbb{R}\). It is obtained a criterion for \(s_0\) to be a root of multiplicity \(2n\) of \(\Delta(s)\). The conditions for \(s_0\) to be a strictly dominant root are proved.
If \(s_0\) is a root of multiplicity \(2n\) of \(\Delta(s)\), then the trivial solution is exponentially stable if \(a_{n-1}>-n^2/\tau\).
Some applications for linear control systems are given.

MSC:

34K20 Stability theory of functional-differential equations
34K06 Linear functional-differential equations
34K35 Control problems for functional-differential equations
93D15 Stabilization of systems by feedback
33C90 Applications of hypergeometric functions

Software:

DLMF; p3delta; Python
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Full Text: DOI arXiv

References:

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