Stability theorems for nonautonomous functional differential equations by Lyapunov functionals
We consider a system of functional differential equations (1) \(x'(t)=F(t,x_ t)\) having finite delay h and satisfying \(F(t,0)=0\). The object is to give conditions on Lyapunov functionals to ensure stability without asking that F(t,\(\phi)\) be bounded for \(\phi\) bounded. We begin by surveying seven classical examples in which we note that an \(L^ 2\)- norm of the solution x(t) frequently appears in the derivative of the Lyapunov functional, but that investigators ignore it in favor of a pointwise norm on x(t). We show that the \(L^ 2\)-norm can be much more useful than the pointwise norm. A measurable function \(\eta\) : \(R\to R\) is said to be integrally positive with parameter \(\delta >0\) (\(\eta\in IP(\delta))\) if whenever \(\{t_ i\}\) and \(\{\delta_ i\}\) satisfy \(t_ i+\delta_ i<t_{i+1}\), \(\delta_ i\geq \delta\), then \(\sum^{\infty}_{i=1}\int^{t_ i+\delta_ i}_{t_ i}\eta (t)dt=\infty.\) Let \(W_ i\) denote a nonnegative increasing function with \(W_ i(0)=0\) and let \(\|| \phi \||\) denote the \(L^ 2\)-norm of \(\phi\). The following is a typical result. Theorem. Let \(\eta \in IP(h_ 1)\) with \(0<h_ 1<h\) and suppose there is a continuous functional V(t,\(\phi)\) satisfying \((i)\quad W_ 1(| \phi (0)|)\leq V(t,\phi)\leq W_ 2(| \phi (0)| +W_ 3(\|| \phi \||)\) and \((ii)\quad V'_{(1)}(t,x_ t)\leq -\eta (t)W_ 4(\|| x_ t\||.\) Then \(x=0\) is asymptotically stable. It is interesting to note that this result fails for ODE's.
scientific article; zbMATH DE number 4109180
scientific article