**Theorem 1.** *Let the following conditions be satisfied:*

*(1) Functions aij*(*t*) *are continuous everywhere except a countable set of points ζ*1, *ζ*2, .., *where the functions have discontinuities. There is at most a finite number of discontinuities on each* [0, *A*], 0 < *A* < ∞*. The coefficients* {*aij*(*t*)} *at ζ*1, *ζ*2, ... , *have discontinuities of the first kind or discontinuities of the second kind integrable in L-metric:*

$$\int\_{\mathbb{Z}\_l^1}^{\mathbb{Z}\_l^1} |a\_{ij}(t)|dt \le c\_{lij} < \infty, i, j = 1, 2, \dots, n, l = 1, 2, \dots$$

*Here, ζ*<sup>1</sup> *<sup>l</sup> are the points that satisfy <sup>ζ</sup><sup>i</sup>* < *<sup>ζ</sup>*<sup>1</sup> *<sup>i</sup>* < *ζi*+1, *i* = 0, 1, . . . , *ζ*<sup>0</sup> = 0;

*(2) The Cauchy problem (10)–(11) has a solution for t* ≥ 0 *and any initial conditions;*

*(3) The inequality* Λ(*A*(*t*)) ≤ −*κ*, *κ* > 0 *is valid everywhere except a set of points ζ*1, *ζ*2, ... *Here, A*(*t*) = {*aij*(*t*)}*<sup>n</sup> <sup>i</sup>*,*j*=1.

*Then, zero solution of the system (10) is asymptotically stable in general.*

Set Λ(*A*(*t*)) equal to zero at discontinuity points *ζi*. By the theorem, we have a finite number of discontinuities in each time interval. Thus, it does not change the value 5 *t* <sup>0</sup> Λ(*A*(*τ*))*dτ*.

**Proof of Theorem 1.** Consider a time interval [0, *ζ*1). The Wintner estimate is valid [47] within this interval

$$\|\mathbf{x}(t)\| \le \|\mathbf{x}(0)\| \exp\left\{ \int\_0^t \Lambda(A(\tau))d\tau \right\}, t \in [0, \mathbb{Z}\_1). \tag{12}$$

The function *x*(*t*) is continuous for *t* ≥ 0, and then the Inequality (12) is correct for *t* ∈ [0, *ζ*1]. Therefore,

$$\|\mathfrak{x}(\zeta\_1)\| \le \|\mathfrak{x}(0)\| \exp\left\{ \int\_0^{\zeta\_1} \Lambda(A(\tau))d\tau \right\}.$$

Consider an interval [*ζ*1, *ζ*2]. First, assume that functions *aij*(*t*) have discontinuities of the first kind at *<sup>ζ</sup>*2. Let *<sup>A</sup>*+(*t*) = {*a*<sup>+</sup> *ij* (*t*)}, *i*, *j* = 1, 2, ... , *n* be a matrix with elements defined by

$$a\_{ij}^+(t) = \begin{cases} a\_{ij}(t), t \neq \mathbb{Q}\_2\\ \lim\_{t \to \mathbb{Q}\_2 + 0} a\_{ij}(t), t = \mathbb{Q}\_2. \end{cases}$$

We take *xi*(*ζ*1), *i* = 1, 2, ... , *n*, as initial values. Repeat the arguments above, for *t* ∈ [*ζ*1, *ζ*2], and we have

$$\begin{aligned} \|\|\mathbf{x}(t)\|\| &\leq \|\mathbf{x}(\zeta\_1)\|\exp\left\{\int\_{\zeta\_1}^t \Lambda(A^+(\tau))d\tau\right\} = \\ &= \|\mathbf{x}(\zeta\_1)\|\exp\left\{\int\_{\zeta\_1}^t \Lambda(A(\tau))d\tau\right\} \leq \|\mathbf{x}(0)\|\exp\left\{\int\_0^t \Lambda(A(\tau))d\tau\right\}. \end{aligned}$$

Next, we consider the case where functions *aij*(*t*) have discontinuities of the second kind and integrals 5 <sup>∞</sup> <sup>0</sup> |*aij*(*t*)|*dt* exist. Obviously, for *t* ∈ [0, *ζ*1), the Inequality (12) is valid. Since the function *x*(*t*) is continuous, the inequality is valid on the interval [0, *ζ*1]. From the continuity of *<sup>x</sup>*(*t*) , it follows that a point *<sup>ζ</sup>*<sup>1</sup> <sup>1</sup>(*ζ*<sup>1</sup> < *<sup>ζ</sup>*<sup>1</sup> <sup>1</sup> < *ζ*2) such that

$$||\mathfrak{x}(t)|| \le ||\mathfrak{x}\_0|| + ||\mathfrak{x}\_0|| \frac{\exp\left\{\int\_0^t \Lambda(A(\tau))d\tau\right\} - 1}{2}$$

exists for *<sup>t</sup>* <sup>∈</sup> [*ζ*1, *<sup>ζ</sup>*<sup>1</sup> 1].

By taking *xi*(*ζ*<sup>1</sup> <sup>1</sup>), *i* = 1, 2, ... , *n* as initial conditions and repeating the arguments above, we verify immediately the validity of the inequality

$$||\mathfrak{x}(t)|| \le ||\mathfrak{x}\_0|| \exp\left\{ \int\_0^t \Lambda(A(\tau)) d\tau \right\},$$

for 0 ≤ *t* ≤ *ζ*2. Repeating the process in each interval [*ζl*, *ζl*+1], we can observe that the inequality is correct for 0 ≤ *t* ≤ ∞.

Let us consider the case when coefficients *aij*(*t*) have a countable number of discontinuity points.

Let the function *a*11(*t*) have a countable number of discontinuity points located in interval [*b*1, *b*2] with measure Δ = |*b*<sup>2</sup> − *b*1|, *b*<sup>1</sup> > 0.

The following assertion is true.

**Theorem 2.** *Let the following conditions be satisfied:*

*(1) The functions aij are continuous everywhere except a countable number of points located in the interval* [*b*1, *b*2], *b*<sup>1</sup> > 0, Δ = |*b*<sup>2</sup> − *b*1|*.*

*(2) The Cauchy problem (10), (11) has a solution for all t* ≥ 0 *and for any initial conditions.*

*(3) The inequality* Λ(*A*(*t*)) ≤ −*κ*, *κ* > 0 *holds everywhere except the interval* [*b*1, *b*2]*. Here, A*(*t*) = {*aij*(*t*)}, *i*, *j* = 1, 2, ..., *n;*

*(4) The inequality is valid*

$$\int\_{0}^{b\_1} \Lambda(A(\tau))d\tau + A\Delta < 0,$$

*where A* = sup *A*(*τ*) *.*

*τ*∈[*b*1,*b*2] *Then, a trivial solution of the system of Equation (10) is asymptotically stable.*

**Proof of Theorem 2.** Consider [0, *b*1]. From Wintner inequality [47], it follows that for *t* ∈ [0, *b*1]

$$\|\|\mathbf{x}(t)\|\| \le \|\mathbf{x}(0)\|\|\exp\left\{\int\_0^t \Lambda(A(\tau))d\tau\right\}.$$

Take [*b*1, *b*2]. Here, we have

$$\mathbf{x}(t) = \mathbf{x}(b\_1) + \int\_{b\_1}^{t} A(\tau)\mathbf{x}(\tau)d\tau.$$

and the integral is understood in the sense of Lebesgue.

Thus,

$$\|\|\mathbf{x}(t)\|\| \le \|\mathbf{x}(b\_1)\| + \int\_{b\_1}^t \|\|A(\tau)\|\| \|\mathbf{x}(\tau)\| \|d\tau \le \|\mathbf{x}(b\_1)\| + A \int\_{b\_1}^t \|\mathbf{x}(\tau)\| \|d\tau.$$

where *<sup>A</sup>* = max*b*1≤*t*≤*b*<sup>2</sup> *<sup>A</sup>*(*t*) .

From Gronwall–Bellman inequality, it follows that

$$\|\|\mathbf{x}(t)\|\| \le \|\mathbf{x}(b\_1)\|\|\exp\{A(t-b\_1)\}.$$

Thus, for *t* ∈ [*b*1, *b*2],

$$\begin{split} \|\mathbf{x}(t)\| &\leq \|\mathbf{x}(0)\| \exp\Big\{\int\_{0}^{b\_{1}} \Lambda(A(\tau))d\tau + A(t - b\_{1})\Big\} \\ &\leq \exp\Big\{\int\_{0}^{b\_{1}} \Lambda(A(\tau))d\tau + A(b\_{2} - b\_{1})\Big\} \|\mathbf{x}(0)\|. \end{split} \tag{13}$$

Therefore, if 5 *<sup>b</sup>*<sup>1</sup> <sup>0</sup> Λ(*A*(*τ*))*dτ* + *A*Δ < 0, then *x*(*b*2) ≤ *x*(0) and a trivial solution of the system of Equation (10) is stable.

It is easy to see that the obtained results can be extended to systems of switching differential equations. At the same time, the stability condition is extended to systems of differential equations with an arbitrary number of relays. Moreover, the suggested method allows one to obtain sufficient conditions for the stability of solutions of systems of nonlinear equations with relay. Similarly, based on the results presented in Sections 2 and 3, one can formulate sufficient conditions for the stability of switching systems.

Example. We consider a system of differential equations with relay

$$\frac{d\mathbf{x}\_k(t)}{dt} = \sum\_{l=1}^m a\_{kl}(t)\operatorname{sgn}\mathbf{x}\_l(t) + \sum\_{l=1}^n b\_{kl}(t)\mathbf{x}\_l(t), \ \ k = 1, 2, \dots, n.$$

Theorem 1 implies that for the asymptotic stability of the trivial solution of this system, it is sufficient to fulfill the following conditions: for each *t* ∈ [0, ∞)

$$|b\_{kk}(t) + \sum\_{l=1}^{m} |a\_{kl}(t)| + \sum\_{l=1, l \neq k}^{n} |b\_{kl}(t)| \le -\xi'\xi > 0.$$

*3.2. Stability of Solutions for Systems of Nonlinear Non-Autonomous Differential Equations with Discontinuous Right-Hand Sides*

First, let us recall the sufficient stability conditions for systems of nonlinear differential equations with continuous right-hand sides that we gave previously [48], and which we extensively use below.

Consider the system of equations

$$\frac{d\mathbf{x}\_i(t)}{dt} = a\_i(t, \mathbf{x}\_1(t), \dots, \mathbf{x}\_n(t)), \ i = 1, 2, \dots, n,\tag{14}$$

with the initial conditions

$$\mathbf{x}\_i(0) = \mathbf{x}\_i, i = 1, 2, \dots, n. \tag{15}$$

Let *x*∗(*t*)=(*x*∗ <sup>1</sup> (*t*), ... , *x*<sup>∗</sup> *<sup>n</sup>*(*t*)) be a steady-state solution of the Cauchy problems (14) and (15).

Let the functions *ai*(*t*, *u*1, ..., *un*) be continuous with respect to the first variable and have partial derivatives with respect to other variables satisfying the Lipschitz condition with a coefficient *A*:

$$|D^j a\_i(t, \mathbf{x}\_1^\*, \dots, \mathbf{x}\_n^\*) - D^j a\_i(t, y\_1^\*, \dots, y\_n^\*)| \le A(|\mathbf{x}\_1^\* - y\_1^\*| + \dots + |\mathbf{x}\_n^\* - y\_n^\*|), \, i, j = 1, \dots, n. \tag{16}$$

Let *χ* = const > 0. Let, for *t* ∈ [0, ∞), the following conditions be satisfied

$$\left| D^{i}a\_{i}(t, \mathbf{x}\_{1}^{\*}(t), \dots, \mathbf{x}\_{n}^{\*}(t)) + \sum\_{j=1, j \neq i}^{n} \left| D^{j}a\_{i}(t, \mathbf{x}\_{1}^{\*}(t), \dots, \mathbf{x}\_{n}^{\*}(t)) \right| < -\chi < 0, \ i = 1, \dots, n. \tag{17}$$

**Theorem 3** ([48])**.** *Let the system (14) have a steady-state solution x*∗(*t*)=(*x*∗ <sup>1</sup> (*t*), ... , *x*<sup>∗</sup> *<sup>n</sup>*(*t*)). *Let the functions ai*(*t*, *x*1, ... , *xn*), *i* = 1, 2, ... , *n be continuous with respect to the first variable, continuously differentiate to other variables and partial derivatives satisfy the Lipschitz condition (16). Let, for all t* ≥ 0*, the conditions (17) be satisfied. Then, the steady-state solution x*∗(*t*) *of the system of Equation (14) is asymptotically stable in the R*<sup>3</sup> *<sup>n</sup> space metric of n-dimensional vectors <sup>v</sup>* = (*v*1,..., *vn*) *with norm <sup>v</sup>* = max1≤*j*≤*<sup>n</sup>* |*vj*|*.*

Consider a system of nonlinear equations

$$\frac{du\_i(t)}{dt} = a\_i(t, u\_1(t), \dots, u\_n(t)), \; i = 1, 2, \dots, n, \; t \ge 0,\tag{18}$$

with initial conditions

$$
\mu\_i(0) = \mu\_{i\prime} \land = 1, 2, \ldots, n. \tag{19}
$$

Their right-hand sides are continuous everywhere except a countable set of values (*ζi*, *u<sup>i</sup>* <sup>1</sup>,..., *<sup>u</sup><sup>i</sup> <sup>n</sup>*), *i* = 1, 2, . . . , in which they have discontinuities.

For the sake of simplicity, we consider two cases here:

(1) there are discontinuities with respect to variable *t*;

(2) there are discontinuities with respect to variable *u*1.

Consider the first case. Assume that the functions *aj*(*t*; *u*1, ... , *un*), *j* = 1, 2, ... , *n* have discontinuities with respect to *t* at points *ζi*, *i* = 1, 2, ... ; 0 < *ζ*<sup>1</sup> < *ζ*<sup>2</sup> < ... . For convenience, let *j* = 1. The functions *ai*(*t*; *u*1,..., *un*), *i* = 2, 3, . . . , *n* are assumed to be continuous.

We impose the following constraints on *ai*(*t*, *u*1,..., *un*), *i* = 1, 2, . . . , *n*:

(1) functions *ai*(*t*, *u*1, ... , *un*), *i* = 2, ... , *n* are continuous with respect to *t*(*t* ∈ [0, ∞)) and have partial derivatives that satisfy the Lipschitz condition with coefficient *q* with respect to other variables

$$|D^k a\_i(t, u\_1, \ldots, u\_n) - D^k a\_i(t, v\_1, \ldots, v\_n)| \le q \sum\_{i=1}^n |u\_i - v\_i|, k = 1, 2, \ldots, n, \ t \in [0, \infty); \quad (20)$$

(2) the function *<sup>a</sup>*1(*t*, *<sup>u</sup>*1, ... , *un*) is continuous for *<sup>t</sup>* <sup>∈</sup> [0, *<sup>ζ</sup>*1) <sup>∪</sup><sup>∞</sup> *<sup>i</sup>*=<sup>1</sup> (*ζi*, *ζi*+1). For *t*, it has partial derivatives with respect to other variables satisfying the Lipschitz condition with coefficient *q*

$$|D^j a\_1(t, u\_1, \dots, u\_n) - D^j a\_1(t, v\_1, \dots, v\_n)| \le q \sum\_{i=1}^n |u\_i - v\_i|\_{\prime \prime}, j = 1, 2, \dots, n;\tag{21}$$

(3) for *t* ∈ [0, ∞)

$$D^j a\_i(t, \mathbf{x}\_1^\*(t), \dots, \mathbf{x}\_n^\*(t)) + \sum\_{j=1, j \neq i}^n \left| D^j a\_i(t, \mathbf{x}\_1^\*(t), \dots, \mathbf{x}\_n^\*(t)) \right| < -\chi < 0, \ i = 2, 3, \dots, n; \tag{22}$$

$$(4)\text{ for }t \in \left[0, \zeta\_1\right) \cup\_{i=1}^{\infty} \left(\zeta\_{i\nu}\zeta\_{i+1}\right).$$

$$\left| D^1 a\_1(t, \mathbf{x}\_1^\*(t), \dots, \mathbf{x}\_n^\*(t)) + \sum\_{j=2}^n \left| D^j a\_1(t, \mathbf{x}\_1^\*(t), \dots, \mathbf{x}\_n^\*(t)) \right| \right| < -\chi < 0,\tag{23}$$

where *χ* = const > 0.

Now, consider the time interval *t* ∈ [0, *ζ*1].

Let *<sup>u</sup>*(0)<sup>≤</sup> *<sup>δ</sup>*, where *<sup>δ</sup>* <sup>≤</sup> *<sup>χ</sup>*/(4*qn*2). In [48], it was shown that in order to fulfill the conditions (20)–(23) in time interval *t* ∈ [0, *ζ*1], the trajectory of the solution of the Cauchy problems (18) and (19) does not leave a ball *B*(0, *δ*). It was also shown that for *t* ∈ [0, *ζ*1], it holds that

$$\|\|u(t)\|\| \le e^{-\chi t/4} \|\|u(0)\|\| \le e^{-\chi t/4} \delta. \tag{24}$$

*k*

Since the function *u*(*t*) is continuous for *t* ∈ [0, ∞), we can find *ζ* <sup>1</sup>, *ζ*<sup>1</sup> < *ζ* <sup>1</sup> < *ζ*<sup>2</sup> such that |*ζ* <sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*1<sup>|</sup> <sup>&</sup>lt; <sup>|</sup>*ζ*<sup>2</sup> <sup>−</sup> *<sup>ζ</sup>*1|/10 and *<sup>u</sup>*(*t*)<sup>≤</sup> *<sup>e</sup>*−*χζ*1/8 *<sup>u</sup>*(0) for *<sup>t</sup>* <sup>∈</sup> [*ζ*1, *<sup>ζ</sup>* <sup>1</sup>]. Obviously, for *t* ∈ [*ζ* <sup>1</sup>, *<sup>ζ</sup>*2], *<sup>u</sup>*(*t*)<sup>≤</sup> *<sup>e</sup>*−*χ*(*t*−*ζ* <sup>1</sup>)/4 *<sup>u</sup>*(*ζ* <sup>1</sup>)<sup>≤</sup> *<sup>e</sup>*−*χ*(*t*−*ζ* <sup>1</sup>)/4*e*−*χζ*1/8 *<sup>u</sup>*(0) .

For *<sup>t</sup>* <sup>=</sup> *<sup>ζ</sup>*2, we have *<sup>u</sup>*(*ζ*2)<sup>≤</sup> *<sup>e</sup>*−*χ*(*ζ*2−*ζ* <sup>1</sup>)/4*e*−*χζ*1/8 *<sup>u</sup>*(0) <sup>&</sup>lt; *<sup>e</sup>*−*χ*Δ1/8*e*−*χ*Δ0/8 *<sup>u</sup>*(0) . Here and below, Δ*<sup>k</sup>* = |*ζk*+<sup>1</sup> − *ζk*|.

Consider the interval [*ζ*2, *ζ*3]. From function *u*(*t*) , *t* ∈ [0, ∞) continuity, it follows that there is an interval [*ζ*2, *ζ* <sup>2</sup>] such that |*ζ* <sup>2</sup> <sup>−</sup> *<sup>ζ</sup>*2<sup>|</sup> <sup>&</sup>lt; <sup>Δ</sup>3/10 and *<sup>u</sup>*(*t*)<sup>≤</sup> *<sup>e</sup>*−*χ*(Δ0+Δ1)/8 *<sup>u</sup>*<sup>0</sup> for *t* ∈ [*ζ*2, *ζ* <sup>2</sup>]. Then, if *t* ∈ [*ζ* <sup>2</sup>, *<sup>ζ</sup>*3] *<sup>u</sup>*(*t*)<sup>≤</sup> *<sup>e</sup>*−*χ*(*t*−*ζ* <sup>2</sup>)/4 *<sup>u</sup>*(*ζ* <sup>2</sup>)<sup>≤</sup> *<sup>e</sup>*−*χ*(*t*−*ζ* <sup>2</sup>)/4*e*−*χ*(Δ0+Δ1)/8 *<sup>u</sup>*<sup>0</sup> , *<sup>u</sup>*(*ζ*3) <sup>&</sup>lt; *<sup>e</sup>*−*χ*(*ζ*3−*ζ*2)/8*e*−*χ*(Δ0+Δ1)/8 *<sup>u</sup>*<sup>0</sup> <sup>=</sup> *<sup>e</sup>*−*χ*(Δ0+Δ1+Δ2)/8 *<sup>u</sup>*<sup>0</sup> .

Repeating the process, we have for *t* ∈ [*ζk*, *ζk*+1] : *u*(*ζk*+1) < *e* −*χ*( ∑ *l*=0 Δ*l*)/8 . Therefore, *t* → ∞ *u*(*t*) → 0.

Asymptotic stability is proven.

## **Theorem 4.** *Let the following conditions be fulfilled:*

*(1) the Cauchy problems (14) and (15) has a steady-state solution x*∗(*t*), *x*∗(*t*)=(*x*∗ <sup>1</sup> (*t*),..., *x*<sup>∗</sup> *<sup>n</sup>*(*t*))*, t* ≥ 0;

*(2) functions aij*(*t*, *x*1, *x*2, ... , *xn*) *are continuous with respect to variables x*1, ... , *xn and have a set of countable discontinuities ζ*1, ... , *ζn*, ... *with respect to t*. *Moreover, in each finite time interval* [0, *T*)*, there is at most a finite number of discontinuities;*

*(3) at every point of continuity with respect to t, functions aij*(*t*, *x*1, ... , *xn*) *have partial derivatives with respect to x*1,..., *xn and satisfy the Inequalities (20), (21);*

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

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

Now, we move on to the case where functions *ai*(*t*, *u*1, ... , *un*), *i* = 1, 2, ... , *n* have discontinuities with respect to *ui*, *i* = 1, 2, . . . , *n*.

For simplicity, we restrict the discussion to the case where the function *a*1(*t*, *u*1, ... , *un*) has a discontinuity at *u*<sup>1</sup> for *u*<sup>1</sup> = *u*<sup>∗</sup> 1.

Assume that a gap occurs at time *t* = *η*<sup>1</sup> > 0 and *u*1(*η*1) = *u*<sup>∗</sup> 1.

Let, for −∞ < *u*<sup>1</sup> < *u*<sup>∗</sup> <sup>1</sup>, −∞ < *ui* < ∞, *i* = 2, ... , *n* and for *u*<sup>∗</sup> <sup>1</sup> < *u*<sup>1</sup> < ∞, −∞ < *ui* < ∞, *i* = 2, ... , *n*, functions *ai*(*t*, *u*1, ... , *un*), *i* = 1, ... , *n* have partial derivatives that satisfy the Lipschitz condition <sup>|</sup>*D<sup>j</sup> ai*(*t*, *<sup>u</sup>*1, ... , *un*) <sup>−</sup> *<sup>D</sup><sup>j</sup> ai*(*t*, *<sup>v</sup>*1, ... , *vn*)| ≤ *<sup>q</sup> <sup>n</sup>* ∑ *i*=1 |*ui* − *vi*|, *j* =

1, 2, . . . , *n*.

Consider a time interval [0, *η*1). The conditions of Theorem 3 are verified in each [0, *<sup>b</sup>*] <sup>⊂</sup> [0, *<sup>η</sup>*1). Therefore, for *<sup>t</sup>* <sup>∈</sup> [0, *<sup>η</sup>*1), the inequality occurs *<sup>u</sup>*(*t*)<sup>≤</sup> *<sup>e</sup>*−*χt*/4*<sup>δ</sup>* <sup>&</sup>lt; *<sup>δ</sup>*.

Although the function *u*(*t*) is continuous for *t* ∈ [0, ∞), the inequality *u*(*t*) ≤ *<sup>e</sup>*−*χt*/4*<sup>δ</sup>* is valid for *<sup>t</sup>* <sup>∈</sup> [0, *<sup>η</sup>*1].

Consider special features of the transition of the Cauchy problem solution trajectories (18) and (19) through *t* = *η*1.

There are two possible cases:

(1) there is an interval (*η*1, *η*<sup>1</sup> + Δ1], in which *u*1(*t*) = *u*1(*η*1);

(2) there is a time interval [*η*1, *η*<sup>1</sup> + Δ2], in which *u*1(*t*) = *u*1(*η*1).

We will study each case separately.

First, from continuity of the function *u*(*t*) , it follows that there is *h*, *h* < *η*1/10 so that *<sup>t</sup>* <sup>∈</sup> [*η*1, *<sup>η</sup>*<sup>1</sup> <sup>+</sup> *<sup>h</sup>*], *<sup>u</sup>*(*t*) <sup>&</sup>lt; *<sup>e</sup>*−*χt*/8*δ*.

Therefore, for *<sup>t</sup>* <sup>∈</sup> [0, *<sup>η</sup>*<sup>1</sup> <sup>+</sup> *<sup>h</sup>*], the inequality *<sup>u</sup>*(*t*)<sup>≤</sup> *<sup>e</sup>*−*χt*/8*<sup>δ</sup>* is valid. Clearly, *<sup>u</sup>*(*η*<sup>1</sup> <sup>+</sup> *<sup>h</sup>*)<sup>≤</sup> *<sup>e</sup>*−*χ*(*η*1+*h*)/8*δ*.

For *<sup>t</sup>* <sup>∈</sup> [*η*<sup>1</sup> <sup>+</sup> *<sup>h</sup>*, <sup>∞</sup>), the inequality *<sup>u</sup>*(*t*)<sup>≤</sup> *<sup>e</sup>*−*χ*(*t*−*η*1−*h*)/4 *<sup>u</sup>*(*η*<sup>1</sup> <sup>+</sup> *<sup>h</sup>*) <sup>≤</sup> *<sup>e</sup>*−*χ*(*t*−*η*1−*h*)/4*e*−*χ*(*η*1+*h*)/8*<sup>δ</sup>* <sup>≤</sup> *<sup>e</sup>*−*χt*/8*<sup>δ</sup>* is valid.

Therefore, *<sup>u</sup>*(*t*)<sup>≤</sup> *<sup>e</sup>*−*χt*/8*δ*, for *<sup>t</sup>* <sup>∈</sup> [0, <sup>∞</sup>).

Thus, for the first case, the stability of the steady-state solution for the system of Equation (18) is proven.

Now, we move on to the second case. Since *u*1(*t*) = *u*1(*η*1) for *t* ∈ [*η*1, *η*<sup>1</sup> + Δ1], in this time interval instead of (18), one should observe the following system of equations

$$0 = a\_1(t, \mu\_1(\eta\_1), \mu\_2(t), \dots, \mu\_n(t)). \tag{25}$$

$$\frac{du\_i(t)}{dt} = a\_i(t, u\_1(\eta\_1), u\_2(t), \dots, u\_n(t)), \ i = 2, \dots, n. \tag{26}$$

The system of Equation (26) is considered under initial condition *ui*(*η*1) = *ui*, *i* = 2, 3, . . . , *n*.

We assume that functions *u*2(*t*), ... , *un*(*t*) satisfy the condition (25) for *t* ∈ [*η*1, *η*<sup>1</sup> + Δ1]. The system of Equation (26) is studied similarly to the system (18) in the space of lower dimension. Sufficient conditions of stability for the solution of the system (26) for *t* ∈ [*η*1, *η*<sup>1</sup> + Δ1] are constructed similarly to the sufficient conditions of stability for the solution of the system (18) on the time interval *t* ∈ [0, *ζ*1]. We omit the details. Finally, we

investigate the time interval *t* ∈ [*η*<sup>1</sup> + Δ1, ∞] and employ the arguments given in [48].

Now, we must consider the case of a countable set of discontinuities. It suffices to observe the case where there are discontinuities with respect to *u*<sup>1</sup> for *u*¯*<sup>i</sup>* <sup>1</sup>, *i* = 1, 2, ... . We suggest that the discontinuities occur at the time moments *t* ∗ *<sup>i</sup>* : *a*1(*t* ∗ *<sup>i</sup>* , *<sup>u</sup>*¯*<sup>i</sup>* 1(*t* ∗ *<sup>i</sup>* ), ... , *un*(*t* ∗ *<sup>i</sup>* )), *i* = 1, 2, ... . As above, we will assume that at each finite time interval [0, *T*], there is a finite number of discontinuities.

Here, we also must consider two cases:

(1) *u<sup>i</sup>* <sup>1</sup>(*t*) <sup>=</sup> *<sup>u</sup>*¯*<sup>i</sup>* 1(*t* ∗ *<sup>i</sup>* ) in an interval *t* ∈ (*t* ∗ *i* , *t* ∗ *<sup>i</sup>* + *<sup>h</sup>*<sup>1</sup> *i* ); (2) there is an interval [*t* ∗ *i* , *t* ∗ *<sup>i</sup>* + *hi*], where *<sup>u</sup><sup>i</sup>* <sup>1</sup>(*t*) = *<sup>u</sup>*¯*<sup>i</sup> i* (*t* ∗ *i* ).

For convenience, we observe the first case. The second one leads to the case of a system of a lower dimension.

To each time moment *t* ∗ *<sup>i</sup>* , *i* = 1, 2, ... associated with a function discontinuity *a*1(*t*, *u*1(*t*), ... , *un*(*t*)) we assign a number *ζ <sup>i</sup>* so that *t* ∗ *<sup>i</sup>* < *ζ <sup>i</sup>* < *t* ∗ *<sup>i</sup>*+1, |*ζ <sup>i</sup>* − *t* ∗ *<sup>i</sup>* |≤|*t* ∗ *<sup>i</sup>*+<sup>1</sup> − *t* ∗ *<sup>i</sup>* |/4, *i* = 1, 2, . . . .

Repeating the above arguments for each time interval [0, *ζ* <sup>1</sup>], [*ζ i* , *ζ <sup>i</sup>*+1], *i* = 1, 2, ··· , we verify the validity of the following statement.
