The boundary value problems for the linear singularly perturbed systems of differential-algebraic equations

Abstract

The article considers the main stages of the development of the theory of asymptotic integration of boundary-value problems for linear singularly perturbed differential-algebraic systems. The need of developing constructive methods of approximate integration of boundary-value problems for differential-algebraic systems is due to the importance of their practical application in the theory of nonlinear oscillations, stability of motion, control theory, radio engineering, and biology.

In the present paper the authors offer a review of literary sources, which consider the methods of constructing asymptotic solutions of singularly perturbed systems of differential equations with a degenerate matrix with derivatives under the condition of stability of the spectrum of the limit pencil of matrices. It is noted that the problem of constructing asymptotic solutions of boundary-value problems for systems of this type is poorly studied, and therefore relevant. In particular, the question of the conditions for the existence and uniqueness of the solutions of these problems and the development of methods for constructing their asymptotics in various cases related to the behavior of the spectrum of the limit pencil of matrices has been poorly researched.