**Theorem 6.** *Let the following conditions be fulfilled:*

*(1) functions ai*, *i* = 1, 2, ... , *n are continuous everywhere in* [0, ∞) *except a finite number of points where they have discontinuities of the first kind;*


*(4) in t* ∈ [0, ∞)*, the following condition is satisfied*

$$\int\_0^t \Lambda(A(\tau))d\tau + \gamma t < 0,$$

*where gamma is defined from the inequality F*(*t*, *x*(*t*)) ≤ *γ x*(*t*) . *Then, the Hopfield neural network is stable in general.*

Consider the case of discontinuity in synapses *wij*(*t*), *i*, *j* = 1, 2, ... , *n*. For convenience, we restrict ourselves to the discontinuity of the function *w*11(*t*) at the time moment *b*1, 0 < *b*<sup>1</sup> < ∞.

Let us represent the system of Equation (27) as

$$\begin{array}{rcl}\frac{d\mathbf{x}\_1(t)}{dt} &=& -a\_1(t)\mathbf{x}\_1(t) + w\_{11}(t)\mathbf{g}\_1'(0)\mathbf{x}\_1(t) \\ &+ w\_{11}(t)u\_1(\mathbf{x}\_1(t)) + \sum\_{j=2}^n w\_{1j}(t)g\_j(\mathbf{x}\_j(t)), \\\\ \mathbf{w}\_{\mathbf{f}^\*}(t) & & \underline{n} \end{array}$$

$$\frac{dx\_i(t)}{dt} = -a\_i(t)x\_i(t) + \sum\_{j=1}^n w\_{ij}(t)g\_j(x\_j(t)), i = 2, 3, \dots, n. \tag{31}$$

Here, *u*1(*x*1(*t*)) = *g*1(*x*1(*t*)) − *g* <sup>1</sup>(0)*x*1(*t*).

It is essential that |*u*1(*x*1(*t*))| = *o*(|*x*1(*t*)|), since we examine the trivial solution of the system (27). Therefore,

$$|\mu\_1(\mathbf{x}\_1(t))| = |\mathbf{g}(\mathbf{x}\_1(t)) - \mathbf{g}\_1(0) - \mathbf{g}\_1'(0)\mathbf{x}\_1(t)| \le B|\mathbf{x}\_1(t)|^2.$$

where *<sup>B</sup>* <sup>=</sup> max <sup>0</sup><*θ*(*x*1(*t*))<<sup>1</sup> |*g*(*θ*(*x*1(*t*)))|.

Obviously, the system of Equation (31) has a structure similar to that of the system of Equation (27). The difference is that the coefficient for *x*1(*t*) now is equal to −*a*1(*t*) + *w*11(*t*)*g* <sup>1</sup>(0), and the vector function *F*(*x*(*t*)) contains *w*11(*t*)*u*1(*x*(*t*)) instead of *w*11(*t*)*g*1(*x*(*t*)).

Taking this remark into account, the assertion of Theorem 7 extends to the system (31).

Finally, we consider the case which involves discontinuous activation functions. Clearly, the system of Equation (18) is a special case of the system of Equation (18). Theorem 5's statements are readily extended to this.
