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 \([t_0,\infty)\), \(b_{jk}\) 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 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)


34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations


Zbl 0724.34060
Full Text: DOI


[1] Hale, J., Theory of Functional Differential Equations (1977), Springer-Verlag
[2] Hale, J.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer-Verlag · Zbl 0787.34002
[3] Čermák, J., On the asymptotic behaviour of solutions of certain functional differential equations, Math. Slovaca, 48, 187-212 (1998) · Zbl 0942.34060
[4] Čermák, J., The asymptotic bounds of solutions of linear delay systems, J. Math. Anal. Appl., 225, 373-388 (1998) · Zbl 0913.34063
[5] Diblı́k, J., Asymptotic representation of solutions of equation \(ẏ(t)=β(t)[y(t)−y(t−τ(t))]\), J. Math. Anal. Appl., 217, 200-215 (1998) · Zbl 0892.34067
[6] Diblı́k, J., A criterion for existence of positive solutions of systems of retarded functional differential equations, Nonlinear Anal., 38, 327-339 (1999) · Zbl 0935.34061
[7] Diblı́k, J., A criterion for convergence of solutions of homogeneous delay linear differential equations, Ann. Polon. Math., LXXII, 115-130 (1999) · Zbl 0953.34065
[8] Khusainov, D., Application of the second Lyapunov method to stability investigation of differential equations with deviations, Arch. Math. (Brno), 34, 127-142 (1998) · Zbl 0915.34066
[9] Ráb, M.; Kalas, J., Stability of dynamical systems in the plane, Differential Integral Equations, 3, 127-144 (1990) · Zbl 0724.34060
[10] Krasovskii, N. N., Certain Problems of the Theory of Stability of Motion (1959), in Russian
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