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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.

34K20Stability theory of functional-differential equations
34K06Linear functional-differential equations
[1]Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[2]Rekhlitskii, Z. I.: On stability of solutions of some linear differential equations with retarded argument in Banach space, Dokl. akad. Nauk SSSR 111, No. 1, 29-32 (1956) · Zbl 0072.33302
[3]Mori, T.; Fukuma, N.; Kuwahara, M.: Simple stability criteria for single and composite linear systems with time delay, Internat. J. Control 34, 1175-1184 (1981) · Zbl 0471.93054 · doi:10.1080/00207178108922590
[4]Horn, R.; Johnson, C.: Matrix theory, (1986)
[5]A.I. Kiryanen, K.V. Galunova, Stability of the equation dx/dt=αx(t-h)+βx(t) with a complex coefficients, in: Partial Differential Equations, Leningrad State Pedagogical University, 1989, pp. 65–72 (in Russian).
[6]Diblík, J.; Svoboda, Z.; Šmarda, Z.: Unstable trivial solution of autonomous differential systems with quadratic right-hand side in a cone, Abstr. appl. Anal. (2010)
[7]A.I. Kiryanen, Stability of the systems with aftereffect with applications, St. Petersburg University, St. Petersburg, 1994 (in Russian). · Zbl 0915.34067
[8]Azbelev, N. V.; Simonov, P. M.: Stability of differential equations with after effect, (2002)
[9]Chen, J.; Latchman, H. A.: Frequency sweeping tests for stability independent of delay, IEEE trans. Automat. control 40, No. 9, 1640-1645 (1995) · Zbl 0834.93044 · doi:10.1109/9.412637