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The stability cone for a delay differential matrix equation. (English) Zbl 1229.34114

Consider the vector differential delay equation

x ˙(t)+Ax(t)+Bx(t-τ)=0,τ>0,(*)

where the matrices A and B admit a simultaneous triangulation. The authors introduce the surface 𝒦 in 3

𝒦:={(u 1 ,u 2 ,u 3 ) 3 :u 1 =-acosω+ωsinω,u 2 =asinω+ωcosω,u 3 =a,-τ<ω<π,a-ω tangω},

which is called the stability cone. They prove the following

Theorem. Let A,B,S m×m , S -1 AS=A T , S -1 BS=B T , where A T and B T are lower triangular matrices with entries λ js , μ js (1j,sm), respectivey. Let the points M j =(u 1j ,u 2j ,u 3j ) (1jm) be defined by

u 1j :=τ|μ jj |cos(argμ jj +τImλ jj ),u 2j :=τ|μ jj |sin(argμ jj +τImλ jj ),u 3j :=τReλ jj ·

Equation (*) is asymptotically stable if and only if all the points M j (1jm) lie inside the stability cone 𝒦. If there exists a point M j lying outside the stability cone, then (*) is unstable.

MSC:
34K20Stability theory of functional-differential equations
34K06Linear functional-differential equations
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