## 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
Full Text:

### References:

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