Criteria for quantitative stability for a class of Razumikhin-type retarded functional differential equations
The practical stability of the Razumikhin-type functional-differential equation \(D^{+}x(t)=f(t,x_{t})\) is discussed, where \(D^{+}\) denotes the upper right-hand derivative, \(f:\mathbb{R} \times C_{r}\to\mathbb{R} ^{n}\), \(C_{r}=C([-r,0],\mathbb{R} ^{n})\), \(r>0\). Denote by \(\mathbb{R} _{+}\) the set of all nonnegative real numbers. A retarded functional-differential equation is called an equation of Razumikhin-type if there exist nonnegative, continuous, strictly increasing, unbounded functions \(\varphi_{1}\), \(\varphi_{2}\), continuous functions \(V:\mathbb{R} ^{n}\times\mathbb{R} \to\mathbb{R} _{+}\), \(W:\mathbb{R} _{+}\times\mathbb{R} \to\mathbb{R} _{+}\), and a continuous nondecreasing function \(p(s)>s\) for \(s>0\) such that: (R1) for all \(t\in\mathbb{R} \), \(x\in\mathbb{R} ^{n}\) \(\varphi_{1}(|x|)\leq V(x,t)\leq\varphi_{2}(|x|)\); (R2) \(D^{+}V(\phi(0),t)\leq-W(|\phi(0)|,t)\) if \(V(\phi(\theta),t+\theta)\leq p(V(\phi(0),t))\) for \(-r\leq\theta\leq 0\). Sufficient conditions for the practical stability are obtained for a subset of such family of equations satisfying additional assumptions. The paper improves some previous results by \textit{Ch. Hou} and \textit{J. Qian} [Automatica 34, No. 7, 921-924 (1998; Zbl 0935.93052); IEEE Trans. Autom. Contr. 43, 958-960 (1998); and J. Math. Anal. Appl. 223, No. 1, 50-61 (1998; Zbl 0908.34060)].
retarded functional-differential equations
scientific article; zbMATH DE number 1375016
scientific article