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Stability and asymptotic behaviour of a two-dimensional differential system with delay. (English) Zbl 1008.34064

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 ,), b jk are locally Lebesgue integrable on [t 0 ,) and the function h satisfies Carathéodory conditions on

[t 0 ,)×{[x 1 ,x 2 ] 2 :x 1 2 +x 2 2 <K}×{[y 1 ,y 2 ] 2 :x 1 2 +x 2 2 <K},

with 0<K. 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 M. Ráb and 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.

Reviewer: J.Diblík (Brno)
34K20Stability theory of functional-differential equations
34K25Asymptotic theory of functional-differential equations
34K12Growth, boundedness, comparison of solutions of functional-differential equations