Stability and asymptotic behaviour of a two-dimensional differential system with delay
The two-dimensional system \[ x'(t)=A(t)x(t)+B(t)x(t-r)+h(t,x(t),x(t-r)) \] is considered, where \(A(t)=(a_{jk}(t))\), \(B(t)=(b_{jk}(t))\), \(j,k=1,2\), are real matrices and \(h(t,x,y)\) is a two-dimensional real vector function. It is supposed that the functions \(a_{jk}\) are absolutely continuous on \([t_0,\infty)\), \(b_{jk}\) are locally Lebesgue integrable on \([t_0,\infty)\) and the function \(h\) satisfies Carathéodory conditions on \[ [t_0,\infty)\times\{[x_1,x_2]\in \mathbb{R}^2: x_1^2+x_2^2<K\}\times \{[y_1,y_2]\in \mathbb{R}^2: x_1^2+x_2^2<K\}, \] with \(0<K\leq\infty\). The authors use an original approach for the investigation -- with the aid of complex variables the system is rewritten into an equivalent equation with complex-valued coefficients. (This idea was used in a previous paper of the first author, too, see also \textit{M. Ráb} and \textit{J. Kalas} [Differ. Integral Equ. 3, No. 1, 127-144 (1990; Zbl 0724.34060)].) Stability and asymptotic stability of the trivial solution, and further asymptotic properties (e.g., the boundedness of all solutions by exponential functions) are studied by means of an appropriate Lyapunov-Krasovskii functional. This approach does not require the uniform stability or uniform asymptotics stability of a corresponding linear system and leads to new, effective and easy applicable results. An illustrative example is considered. The authors discuss possible generalizations, too.
scientific article; zbMATH DE number 1764683
scientific article