Stability criteria of differential systems via Lyapunov's second method
The author considers the following differential equations: \[ x'= f(t,x),\tag{1} \] \[ x'= f(t,x)+ h(t,x),\tag{2} \] with \(f,h: J\times \mathbb{R}^n\to \mathbb{R}^n\), \(J= [t_0,\infty)\), \(f(t,0)= h(t,0)= 0\), \(f\), \(h\) are continuously differentiable functions with respect to the components of \(x\) on \(J\times \mathbb{R}^n\) for \(t\in J\), and \[ x'= F(t,x_t),\tag{3} \] \[ x'= F(t, x_t)+ G(t,x_t),\tag{4} \] with \(F,G:\mathbb{R}\times C([-r,0],\mathbb{R}^n)\to \mathbb{R}^n\), \(F(t,0)- G(t,0)= 0\), \(F(t,x_t)\) and \(G(t, x_t)\) are continuous functionals on \(\mathbb{R}\times C\), \(x_t= x(t+\theta,\varphi)\) with the initial value \(x_{t0}= \varphi\), \(\theta\in [-r,0]\), \(r>0\), \(\mathbb{R}\) is the real line and \(C\) denotes the Banach space of continuous functions \(\varphi: [-r,0]\to \mathbb{R}^n\), \(\varphi\in C\) with the norm \(\|\varphi\|= \sup|\varphi(\theta)|\). For these equations the author studies sufficient conditions of different types of stability (uniform stability; Lipschitz stability; exponential stability). Moreover, the author improves some results of \textit{Z. S. Athanassov} [J. Math. Anal. Appl. 86, 194-207 (1982; Zbl 0508.34036)], \textit{T. Yoneyama} [J. Math. Anal. Appl. 125, 161-173 (1987; Zbl 0655.34062)] and \textit{Y. Fu} [Pac. J. Math. 151, No. 2, 229-235 (1991; Zbl 0744.34069)]. Furthermore, he shows that the condition \(V'(t,0)\leq 0\) is not sometimes a necessary condition for the stability results.
scientific article; zbMATH DE number 1311211