Lagrange Stability of Semilinear
Differential-Algebraic Equations and
Application to Nonlinear Electrical Circuits

Maria S. Filipkovska

B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: filipkovskaya@ilt.kharkov.ua

Received January 12, 2017.

Abstract

A semilinear differential-algebraic equation (DAE) is studied focusing
on the Lagrange stability (instability). The conditions for the existence and
uniqueness of global solutions (a solution exists on an infinite interval) of the Cauchy problem, as well as the conditions of the boundedness of the
global solutions, are obtained. Furthermore, the obtained conditions of the Lagrange stability of the semilinear DAE guarantee that every its solution
is global and bounded and, in contrast to the theorems on the Lyapunov
stability, allow us to prove the existence and uniqueness of global solutions regardless of the presence and the number of equilibrium points. We also obtain the conditions for the existence and uniqueness of solutions with a finite escape time (a solution exists on a finite interval and is unbounded, i.e., is Lagrange unstable) for the Cauchy problem. The constraints of the type of global Lipschitz condition are not used which allows to apply efficiently the work results for solving practical problems. The mathematical model of a radio engineering filter with nonlinear elements is studied as an application.
The numerical analysis of the model verifies theoretical studies.