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Critical delays and polynomial eigenvalue problems. (English) Zbl 1166.65040
The author proposes a method for parameterizing the critical delays, \(h_k\), which give the boundary of the stability region of the delay-differential equation \(x'(t)=A_0x(t)+\sum _{k=1}^m A_kx(t-h_k)\) for \(t>0\), with \(x(t)=\varphi (t)\) for \(t\in [-h_m,0]\), where the \(A_k\) are real \(n\times n\) matrices. A key step in the method requires the solution of a quadratic eigenvalue problem for \(n^2\times n^2\) matrices. In the case in which the delays are integer multiples of the smallest delay, the author discusses the relationship of this work to earlier work of J. Chen, G. Gu and C. N. Nett [Syst. Control Lett. 26, No. 2, 107–117 (1995; Zbl 0877.93117)].

MSC:
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34K20 Stability theory of functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
93C23 Control/observation systems governed by functional-differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations
93B35 Sensitivity (robustness)
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