Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks

Ilya Boykov, Vladimir Roudnev and Alla Boykova
Additional contact information
Ilya Boykov: Department of Higher and Applied Mathematics, Penza State University, 40, Krasnaya Str., 440026 Penza, Russia
Vladimir Roudnev: Department of Computational Physics, Saint Petersburg State University, 1 Ulyanovskaya Str., 198504 Saint Petersburg, Russia
Alla Boykova: Department of Higher and Applied Mathematics, Penza State University, 40, Krasnaya Str., 440026 Penza, Russia

Mathematics, 2022, vol. 10, issue 9, 1-16

Abstract: In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. Stability sufficient conditions for nonlinear equations are expressed as the logarithmic norm of the Jacobian of the right-hand side of the system of equations. Sufficient conditions for the stability of solutions of systems of differential equations expressed in terms of logarithmic norms of the right-hand sides of equations (for systems of linear equations) and the Jacobian of right-hand sides (for nonlinear equations) have the following advantages: (1) in investigating stability in different metrics from the same standpoints, we have obtained a set of sufficient conditions; (2) sufficient conditions are easily expressed; (3) robustness areas of systems are easily determined with respect to the variation of their parameters; (4) in case of impulse action, information on moments of impact distribution is not required; (5) a method to obtain sufficient conditions of stability is extended to other definitions of stability (in particular, to p-moment stability). The obtained sufficient conditions are used to study Hopfield neural networks with discontinuous synapses and discontinuous activation functions.

Keywords: differential equations with discontinuous right-hand sides; Hopfield artificial neural networks; stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/9/1524/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/9/1524/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:9:p:1524-:d:807516

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().