Lagrange Stability and Instability of Irregular Semilinear Differential-Algebraic Equations and Applications

We consider an irregular ( singular) semilinear differential- algebraic equation d dt [ Ax] + Bx = f( t, x) and prove the theorems on Lagrange stability and instability. These theorems give sufficient conditions for the existence, uniqueness, and boundedness of the global solution to the Cauchy problem for a 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 the solution of more general classes of applied problems. Two mathematical models of radio- engineering filters with nonlinear elements are studied as applications.