# Hatvani László honlapja

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 Cím: On the asymptotic stability by Lyapunov functionals with semidefinite derivatives. Szerzõ: Hatvani, L. Forrás: Nonlinear Anal., Theory Methods Appl. 30, No.8, 4713-4721 (1997). Nyelv: English Absztrakt: Two stability problems for nonautonomous differential systems are investigated. In the first part, sufficient conditions are given for the asymptotic stability of the zero solution of nonautonomous functional-differential equations (FDE's) with finite delay, of the form $$x'(t)=F(t,x_t), \quad F(t,0) \equiv 0.$$ $F$ is continuous and maps bounded sets into bounded sets. In the second part the problem of stabilization (to asymptotic stability) of the equilibrium of a nonlinear oscillator by intermittent damping is considered. The oscillator is governed by a second order nonlinear ordinary differential equation (ODE) $$x'' +h(t,x,x')x'+f(x)=0.$$ The main features of the two problems making reasonable a common treatment are the following:\par a) The equations in both models are nonautonomous, even strongly nonautonomous, in the sense that the time functions (the varying parameters, coefficients) involved in the equations are not supposed to be periodic or to have any tendency at $t\to \infty$. Consequently, such methods as the invariance principle, limiting equations, etc. cannot be applied. What is more, the time functions in the equations (e.g. the damping coefficient) may be unbounded.\par b) The stability effects in the systems are controlled only over an infinite sequence of time intervals (e.g. intermittent damping).\par c) The derivative of the Lyapunov functionals with respect to the equations are not negative definite, they are only negative semidefinite.} \RV{V.Dragan (Bucure\c{s}ti) Letöltés: | Zentralblatt