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

Consider the vector differential delay equation

$\stackrel{˙}{x}\left(t\right)+Ax\left(t\right)+Bx\left(t-\tau \right)=0,\phantom{\rule{1.em}{0ex}}\tau >0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where the matrices $A$ and $B$ admit a simultaneous triangulation. The authors introduce the surface $𝒦$ in ${ℝ}^{3}$

$\begin{array}{c}𝒦:=\left\{\left({u}_{1},{u}_{2},{u}_{3}\right)\in {ℝ}^{3}:{u}_{1}=-acos\omega +\omega sin\omega ,\hfill \\ \hfill {u}_{2}=asin\omega +\omega cos\omega ,\phantom{\rule{4pt}{0ex}}{u}_{3}=a,\phantom{\rule{4pt}{0ex}}-\tau <\omega <\pi ,a\ge -\frac{\omega }{tang\omega }\right\},\end{array}$

which is called the stability cone. They prove the following

Theorem. Let $A,B,S\in {ℝ}^{m×m}$, ${S}^{-1}AS={A}_{T}$, ${S}^{-1}BS={B}_{T}$, where ${A}_{T}$ and ${B}_{T}$ are lower triangular matrices with entries ${\lambda }_{js}$, ${\mu }_{js}$ $\left(1\le j,s\le m\right)$, respectivey. Let the points ${M}_{j}=\left({u}_{1j},{u}_{2j},{u}_{3j}\right)$ $\left(1\le j\le m\right)$ be defined by

$\begin{array}{cc}\hfill {u}_{1j}& :=\tau |{\mu }_{jj}|cos\left(arg{\mu }_{jj}+\tau \phantom{\rule{0.166667em}{0ex}}\text{Im}\phantom{\rule{0.166667em}{0ex}}{\lambda }_{jj}\right),\hfill \\ \hfill {u}_{2j}& :=\tau |{\mu }_{jj}|sin\left(arg{\mu }_{jj}+\tau \phantom{\rule{0.166667em}{0ex}}\text{Im}\phantom{\rule{0.166667em}{0ex}}{\lambda }_{jj}\right),\hfill \\ \hfill {u}_{3j}& :=\tau \phantom{\rule{0.166667em}{0ex}}\text{Re}\phantom{\rule{0.166667em}{0ex}}{\lambda }_{jj}·\hfill \end{array}$

Equation $\left(*\right)$ is asymptotically stable if and only if all the points ${M}_{j}$ $\left(1\le j\le m\right)$ lie inside the stability cone $𝒦$. If there exists a point ${M}_{j}$ lying outside the stability cone, then $\left(*\right)$ is unstable.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K06 Linear functional-differential equations
##### Keywords:
stability; delay equation; simultaneous triangularization
##### References:
 [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=$\alpha$x(t-h)+$\beta$x(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