On the asymptotic stability in functional differential equations

Consider a system of functional differential equations $dx/dt=f(t,x_{t})$ where $f$ is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional $V(t,\varphi )$ and negative definite functional ${dV}/{dt}$. In applications one can construct a positive definite functional $V$, whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional $f$ in functional differential equations is autonomous ($f$ does not depend on $t$), and N. N. Krasovskii created such criterion for the case where the functional $f$ is periodic in $t$. For the general case of the non-autonomous functional $f$ V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when $f$ is almost periodic in $t$. This case is a particular case of the class of non-autonomous functionals.