**4. Stability of Hopfield Neural Networks**

In this section, we investigate the stability of Hopfield neural networks, which are modeled by a system of nonlinear differential equations

$$\frac{d\mathbf{x}\_i(t)}{dt} = -a\_i(t)\mathbf{x}\_i(t) + \sum\_{j=1}^n w\_{i,j}(t)g\_j(\mathbf{x}\_j(t)),\tag{27}$$

with discontinuous coefficients *ai*(*t*) and activation functions *gi*(*x*), *i* = 1, 2, . . . , *n*.

We will perform our study of the stability of neural networks (27) in two stages. The first stage includes a case with discontinuous coefficients *ai*(*t*). The second one considers discontinuity of activation function *gi*(*x*), *i* = 1, 2, . . . , *n*.

First, let functions *ai*(*t*), *i* = 1, 2, ... , *n* have discontinuities of the first kind. It is enough to restrict ourselves to the case of one point of discontinuity. Assume that the function *a*11(*t*) is discontinuous at the point *b*1, 0 < *b*<sup>1</sup> < ∞. Without loss of generality, we suggest *gj*(0) = 0, *j* = 1, 2, . . . , *n*, |*gi*(*x*)| ≤ *αi*|*x*|, *i* = 1, 2, . . . , *n*.

Now, we investigate the stability of the zero solution of the system of Equation (27). In the interval (0, *b*1], the norm of the solution of Equation (27) for initial value

$$\mathbf{x}(0) = \mathbf{x}\_0.\mathbf{x}(0) = (\mathbf{x}\_1(0), \dots, \mathbf{x}\_n(0)),\tag{28}$$

is estimated by the inequality

$$\begin{split} \|\mathbf{x}(t)\| &\leq \exp\left\{\int\_{0}^{t} \Lambda(A(\tau))d\tau\right\} \|\mathbf{x}(0)\| \\ &+ \int\_{0}^{t} \exp\left\{\int\_{s}^{t} \Lambda(A(\tau))d\tau\right\} \|F(t, \mathbf{x}(s))\| \,\|d\mathbf{s}\_{\prime} \end{split} \tag{29}$$

where *A*(*t*) = {*aij*(*t*)}, *i*, *j* = 1, 2, . . . , *n*,

$$F(t, \mathbf{x}(t)) = \left(\sum\_{j=1}^{n} w\_{1j}(t)\mathbf{g}\_j(\mathbf{x}(t)), \dots, \sum\_{j=1}^{n} w\_{nj}(t)\mathbf{g}\_j(\mathbf{x}(t))\right)^T.$$

Proceeding with the Inequality (29), we have

$$\|\mathbf{x}(t)\| \le \exp\left\{\int\_0^t \Lambda(A(\tau))d\tau\right\} \|\mathbf{x}(0)\| + \gamma \int\_0^t \exp\left\{\int\_s^t \Lambda(A(\tau))d\tau\right\} \|\mathbf{x}(s)\| ds,\tag{30}$$

where *γ* is defined from the inequality *F*(*t*, *x*(*t*)) ≤ *γ x*(*t*) .

From the Inequality (30), using well-known methods, we have the estimate

$$\|\|x(t)\|\| \le \exp\left\{ \int\_0^t \Lambda(A(\tau))d\tau + \gamma t \right\} \|\|x(0)\|\|, t \in [0, b\_1].$$

Taking *x*(*b*1)=(*x*1(*b*1), ... , *xn*(*b*1)) as the initial value and repeating the arguments given in the proof of Theorem 1, we obtain the inequality

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

which is valid for *t* ∈ [0, ∞).

From this inequality, it follows that when the condition

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

is satisfied, the system of Equation (27) is asymptotically stable in general. Thus, the following statement has been proven.
