On the uniform stability of a system of differential equations with complex coefficients (Q1263723)

The author considers the homogeneous linear ordinary differential system \(x'=Ax\), where A is an \(n\times n\) complex constant matrix and x(t) is a column vector of dimension n, \(2\leq n\leq 4\). With the help of Hurwitz polynomials (the polynomial with complex coefficients is a Hurwitz polynomial if all its roots have negative real parts) and positive functions (a function h(\(\lambda)\) is said to be positive if Re h(t)\(>0\) when Re \(\lambda\) \(>0)\), the author extends the results of \textit{Z. Zahreddine} and \textit{E. F. Elshehawey} [Indian J. Pure Appl. Math. 19, No.10, 963-972 (1988; Zbl 0667.34070)] and establishes necessary and sufficient conditions for the stability and uniform stability of the system. For instance, in the case when A is a \(2\times 2\) complex matrix with characteristic polynomial \(f(\lambda)=\lambda^ 2+a_ 1\lambda +a_ 2\) and no repeated zero eigenvalue, the author proves that the system is stable if and only if one of the following two conditions holds: 1) Re \(a_ 1>0\), Re \(a_ 1\cdot Re(a_ 1\bar a_ 2)-(Im a_ 2)^ 2\geq 0\); 2) Re \(a_ 1=Im a_ 2=0\) and \(a^ 2_ 1-4a_ 2\leq 0\), where asymptotic stability occurs only when both inequalities in 1) are strict.