Lagrange stability and instability of nonregular semilinear differential-algebraic equations and applications

Abstract

We consider an nonregular (singular) semilinear differential-algebraic equation $$\frac d{dt} [Ax] + Bx = f(t, x)$$ and prove the theorems on Lagrange stability and instability. The theorems give sufficient conditions for the existence, uniqueness, and boundedness of a global solution of the Cauchy problem for the semilinear differential-algebraic equation and sufficient conditions for the existence and uniqueness of the solution with finite escape time for the analyzed Cauchy problem (this solution is defined on a finite interval and unbounded). The proposed theorems do not contain constraints similar to the global Lipschitz condition. This enables us to use them for solving more general classes of applied problems. Two mathematical models of radioengineering filters with nonlinear elements are studied as applications.