On stability of a class of quasilinear systems with constant delay
A quasilinear system of ordinary differential equations with constant delay of the form \[ \frac{dx(t)}{dt} = e^{\nu t} [A(t) x(t) + B(t) x(t-\tau) + f(t, x(t), x(t-\tau))], \] \[ t \geq t_0 \geq t_*, \quad \tau = const > 0, \quad \nu = const > 0, \] is considered. Here \(A(t)\) and \(B(t)\) are continuous matrix-valued functions. The nonlinear vector-function \(f(t,u,v)\) satisfies the Lipschitz-type condition \[ |e^{\nu t} [f(t, u_1, v_1) - f(t,u_2,v_2) ]|\leq e^{\nu t} \psi(t) [|u_1 - u_2 |+ |v_1 - v_2 |], \] where \(\psi(t)\) is a decreasing positive function, \(\int_{t_*}^{\infty} \psi(t) dt < \infty\). The author proves that stability of a solution to this system follows from stability to the solution of the linear system which can be obtained from the original system by putting \(f = 0\).
scientific article; zbMATH DE number 1046574