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**Delay-dependent stability of high-order neutral systems.**
*(English)*
Zbl 1499.34375

The authors consider differential systems of neutral type with large delay. They generalize the theory to higher order equations as well as to systems of the first order equations. They define an eigenvalue of the neutral delay system as the root of the characteristic function and distinguish between stable eigenvalues and unstable ones. Based on the eigenvalues, they also give the conditions for the neutral delay system to be asymptotically stable. In the theoretical sense, the stability criteria for the first-order neutral delay system can be directly applied to the \(n\)-th order neutral delay system. However, since the parameter matrices of the \(n\)-th order neutral delay system are of large size, much computational effort is needed to apply the same criterion to the large problems. The authors formulate theorem for a bound for unstable eigenvalues and give the full proof of it. Using the argument principle, two theorems concerning the number of unstable eigenvalues of the system are derived by the spectral radius of a nonnegative matrix. The nonnegative matrix is related to the coefficient matrices. The authors give a stability criterion for these systems and use this criterion to obtain a numerical algorithm. The algorithm avoids the computation of the coefficients of the characteristic function. Instead it evaluates the determinant of numerical matrix through the elementary row (or column) operations which are relatively efficient ways. A scalar stability test needs information of all the coefficients of the characteristic function. For large problems the computation of the coefficients of the characteristic function is an ill-posed problem, even if the characteristic function is a polynomial. Although we can obtain the coefficients, this cannot work well in practice for large problems. The authors conclude the paper with a particular numerical example.

Reviewer: Drahoslava Janovská (Praha)

### MSC:

34K20 | Stability theory of functional-differential equations |

34K06 | Linear functional-differential equations |

34K40 | Neutral functional-differential equations |

34K08 | Spectral theory of functional-differential operators |