Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay. (English) Zbl 1212.34235

Rebenda, J.

Arch. Math., Brno 45, No. 3, 223-236 (2009).

Summary: Stability and asymptotic properties of solutions of the real two-dimensional system \[ x'(t) = \mathbf A (t) x(t) + \mathbf B (t) x (\tau (t)) + \mathbf h (t, x(t), x(\tau (t))) \] are studied, where \(\mathbf A\), \(\mathbf B\) are matrix functions, \(\mathbf h\) is a vector function and \(\tau (t) \leq t\) is a nonconstant delay which is absolutely continuous and satisfies \(\lim _{t \to \infty } \tau (t) = \infty \). Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained by using the methods of complexification and Lyapunov-Krasovskii functional.

MSC:

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

Keywords:

stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method