Stability for functional differential equations and some variational problems (Q910920)

The application of Razumikhin type stability results for functional differential equations leads to certain nonstandard variational problems. These problems are posed in this paper. A typical one is the following: for fixed \(T>0\) and given sequences \(\{b_ i\}\), \(\{c_ i\}\) of nonnegative numbers find or approximate sup\(\{\) f(t): \(f\in C^{\infty}([0,T],R)\), \(f(0)=0\), \(-b_ i\leq f^{(i)}(t)\leq c_ i\), \(i=0,1,...\}\), \(0\leq t\leq T\). Only the particular case \(T=\pi /2\), \(b_ i=c_ i=1\) is solved in this paper: \(\sup_ ff(t)=\sin t\), \(0\leq t\leq \pi /2\). Applying this result, it is shown that Barnea's method [\textit{D. I. Barnea}, SIAM J. Appl. Math. 17, 681-697 (1969; Zbl 0181.101)] can be used to get sharp stability results for linear scalar functional differential equations. Especially, if \(\int^{0}_{-r}| s| d\mu (s)\leq \pi /2\) and \(\mu\) is nondecreasing, then the zero solution of \(x'(t)=-\int^{0}_{-r}x(t+s)d\mu (s)\) is stable.