**Theorem 5.** *Let the following conditions be fulfilled:*

*(1) the Cauchy problems (14)–(15) has a steady-state solution x*∗(*t*), *x*∗(*t*)=(*x*∗ <sup>1</sup> (*t*),..., *x*<sup>∗</sup> *<sup>n</sup>*(*t*)); *(2) functions aij*(*t*, *x*1, ... , *xn*) *are continuous with respect to variables* (*t*, *x*1, ... , *xn*) *everywhere except a countable set of discontinuities with respect to variables x*1, ... , *xn that occur at time moments t* ∗ *<sup>i</sup>* , *i* = 1, 2, ... . *Moreover, in each finite time interval, there is a finite number of discontinuities;*

*(3) at continuity points, functions aij*(*t*, *x*1, ... , *xn*) *have partial derivatives with respect to variables x*1,..., *xn that satisfy the Lipschitz condition;*

*(4) the conditions (22), (23) are fulfilled.*

*Then, a steady-state solution of the Cauchy problems (14)–(15) is asymptotically stable.*
