## 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)

### MSC:

 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:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.