Stability criterion for linear systems with nonlinear delayed perturbations
An autonomous linear system with nonlinear perturbation and retarded argument \[ x'(t)= Ax(t)+ f(t,x(t- h_1(t)),\dots, x(t- h_n(t))),\quad x\in\mathbb{R}^n,\tag{\(*\)} \] is considered, with \(A\in\mathbb{R}^{n\times n}\), \(0\leq h_i(t)\leq\overline h=\text{const}\), and \(f\) is sublinear. The matrix \(A\) is called quasi-dominant if there exist \(d_i>0\) such that \(d_i a_{ii}> \sum_{j\neq i} d_i|a_{ij}|\). The authors connect with \(A\) a quasi-dominant matrix \(M\). When \(M\) is quasi-dominant then \((*)\) is globally asymptotically stable.
globally asymptotically stable solutions
scientific article; zbMATH DE number 1346377
scientific article