Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits

TL;DR

This paper investigates the stability and solution properties of semilinear differential-algebraic equations, providing new conditions for global existence, boundedness, and instability, with applications to nonlinear electrical circuits.

Contribution

It introduces novel criteria for Lagrange stability and instability of semilinear DAEs without relying on global Lipschitz conditions, applicable to practical engineering models.

Findings

Established conditions for existence and uniqueness of global solutions.

Derived criteria for boundedness and Lagrange stability.

Validated results through numerical analysis of a nonlinear radio engineering filter.

Abstract

We study a semilinear differential-algebraic equation (DAE) with the focus 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 conditions of the boundedness of the global solutions, are obtained. Furthermore, the obtained conditions for the Lagrange stability of the semilinear DAE guarantee that every its solution is global and bounded, and, in contrast to theorems on the Lyapunov stability, allow to prove the existence and uniqueness of global solutions regardless of the presence and the number of equilibrium points. We also obtain the conditions of 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. We do not use constraints of a global Lipschitz condition type, that allows to use the work results efficiently in practical applications. The mathematical model of a radio engineering filter with nonlinear elements is studied as an application. The numerical analysis of the model verifies the results of theoretical investigations.