TY - JOUR

T1 - Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks

AU - Boykov, Ilya

AU - Roudnev, Vladimir

AU - Boykova, Alla

PY - 2022/5/1

Y1 - 2022/5/1

N2 - 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.

AB - 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.

KW - differential equations with discontinuous right-hand sides

KW - Hopfield artificial neural networks

KW - stability

UR - http://www.scopus.com/inward/record.url?scp=85129792463&partnerID=8YFLogxK

U2 - 10.3390/math10091524

DO - 10.3390/math10091524

M3 - Article

AN - SCOPUS:85129792463

VL - 10

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 9

M1 - 1524

ER -