On Lyapunov's direct method for nonautonomous functional differential equations (Q2784704)

The paper has been written by a prominent scientist in the area of functional-differential equations. A nonautonomous system of functional-differential equations with delay NEWLINE\[NEWLINE \frac{dx(t)}{dt}=F(t,x_t),\quad F(t,0)\equiv 0,\tag{1}NEWLINE\]NEWLINE is considered, and the asymptotic stability of its solution NEWLINE\[NEWLINE x(t)\equiv 0\tag{2}NEWLINE\]NEWLINE is studied. Sufficient conditions for asymptotic stability and uniform asymptotic stability of solution (2) to equation (1) with finite delays are formulated by the method of Lyapunov functionals. The derivatives of the functionals with respect to the equations are negative semidefinite in terms of either \(|x(t)|\) or \(L_2\)-norm of segment \(x_t\) and may depend explicitly on time \(t\). The theorems do not require the boundedness of the right-hand sides in the equations.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00016].