**5. Conclusions**

In this paper, we obtain sufficient conditions of asymptotic stability for solutions of linear and nonlinear systems of ordinary differential equations with discontinuous righthand sides. We have derived conditions for local asymptotic stability and stability in general and the obtained sufficient conditions have been used to investigate the stability of Hopfield neural networks with discontinuous synapses and activation functions. The proposed method for studying Hopfield neural networks can also be applied to other types of artificial neural networks.

The authors hope to continue their study in the following directions:


We intend to use the obtained results in the following fields:


**Author Contributions:** I.B. and A.B. provided sufficient conditions for the stability of systems of differential equations with discontinuous right-hand sides. V.R. obtained stability conditions for Hopfield neural networks. A.B. reviewed the literature. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest
