A note on exponential stability of quasi-linear ordinary differential equations (Q5938930)

Here, the system of quasilinear ordinary differential equations NEWLINE\[NEWLINEdx/dt= A(t)x+g(t,x) \tag{1}NEWLINE\]NEWLINE is considered, where \(A:[0,\infty) \to\mathbb{C}^{n \times n}\), \(g:[0,\infty) \times B_\eta(0) \to\mathbb{C}^n\) are continuous functions and \(B_\eta(0)= \{x\in\mathbb{C}^n \mid x<\eta\}\) for \(\eta>0\); \(\mathbb{C}^n\) is the \(n\)-dimensional space of complex column vectors. It is assumed that \(g(t,x)\) satisfies the condition of quasilinearity \(\lim_{x\to 0}{|g(t,x) |\over |x|}=0\) uniformly in \(t\), \(0\leq t<\infty\).NEWLINENEWLINENEWLINEIt is proved that, under this condition, the exponential stability of the zero solution to the linear equation \(dx/dt= A(t)x\) is not only sufficient, but also necessary for the exponential stability of the zero solution to the perturbed equation (1).NEWLINENEWLINENEWLINEThe authors study also the exponential stability of a given solution to the nonlinear system \(dx/dtf(t,x)\), where \(f:[0,\infty) \otimes\mathbb{C}^n \to\mathbb{C}^n\) is a continuous function.