A note on exponential stability of quasi-linear ordinary differential equations (Q6205167)

The exponential stability of the zero solution to the system of nonlinear ordinary differential equations \[ {dx\over dt}= f(t,x)\tag{1} \] with \(f\) continuous on \([0,\infty)\times \mathbb{C}^n\) is considered. If (2) \(f(t, x)= A(t) x+ g(t,x)\) and \[ \lim_{x\to 0} {|g(t,x)|\over|x|}= 0\quad\text{uniformly in }t,\quad t\geq 0,\tag{3} \] where \(A: [0,\infty)\to \mathbb{C}_{n\times n}\) and \(g: [0,\infty)\times \mathbb{C}^n\) are continuous, then system (1) is called quasilinear. It is proved (under some additional assumptions) that the exponential stability of the zero solution to the quasilinear system of differential equations (1) can be completely characterized by the exponential stability of the zero solution to the system of linear differential equations \[ {dx\over dt}= A(t)x.\tag{4} \] Some important special cases for periodic \(f\) as well as for autonomous equations are discussed.