Equivalent conditions on stability of functional differential equations
The paper deals with stability conditions (in various senses) for the equations \(x'(t)=F(t,X_ t)\), where \(X_ t(s)=X(t+s)\), \(-h\leq s\leq 0\), by means of Lyapunov's second method. After a brief review of the known results in this regard (Krasovskij, Burton, Hatvani), the author provides some improvements. The main concern is related to the equivalence of two types of conditions intervening on the ``wedge'' appearing in the limitation of the derivative of a Lyapunov functional \(V(t,\varphi)\) with respect to the system. Namely, usual wedges for systems with delay render the same services as the wedges of the form \(W\Bigl(\int^ t_{t- h}| X(s)| ds\Bigr)\). Several examples are considered, related to special equations, such as \(x'(t)=b(t)f(x(t-h))\), \(x'(t)=- a(t)x(t)+b(t)\int^ t_{t-h} x(u)du\).
scientific article; zbMATH DE number 150644