Oscillation and nonoscillation properties of two-dimensional delay direct control systems
The author considers a delay direct control system (1) \(x'(t)+Ax(t)+Bx(t- \tau)=hf(\sigma)\), \(t\geq 0\), where \(x(t)=(x_ 1(t),x_ 2(t))^ T\), \(x(t-\tau)=(x_ 1(t-\tau_ 1),x_ 2(t-\tau_ 2))^ T\), \(A\), \(B\) are constant \(2\times 2\) matrices, \(h=(h_ 1,h_ 2)^ T\), \(\sigma=Cx(t)\), \(C=(c_ 1,c_ 2)\), \(\tau_ i(0)\), \(h_ i,c_ i\), \(i=1,2\), are constants, \(f\) is a scalar continuous function such that \(f(0)=0\), \(0<f(\sigma)/\sigma<k\) for \(\sigma\neq 0\), \(k\)-const. The author gives sufficient conditions under which all bounded solutions of (1) are i) oscillatory or every component tends monotonically to zero as \(t\to\infty\); ii) oscillatory.
scientific article; zbMATH DE number 123636