Projector Approach to Constructing Asymptotic Solution of Initial Value Problems for Singularly Perturbed Systems in Critical Case

Galina Kurina

doi:10.3390/axioms8020056

Abstract

Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix A ( t ) standing in front of the unknown function. In the present paper, the orthogonal projectors onto k e r A ( t ) and k e r A ( t ) ′ (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction.

Highlights

IntroductionThe bibliography of publications devoted to singularly perturbed problems is very extensive
The bibliography of publications devoted to singularly perturbed problems is very extensive.Most of them deal with problems in which a degenerate equation, following from the original one where a small parameter is equal to zero, is resolvable with respect to a fast component of an unknown variable
It allows us to represent the algorithm of the boundary functions method for constructing an asymptotic solution of initial-value singularly perturbed problems in the critical case more clearly than in [4]

Summary

Introduction

The bibliography of publications devoted to singularly perturbed problems is very extensive. An asymptotic solution containing boundary functions for the initial value problem of the weakly non-linear differential equation in a real m-dimensional space X:. It allows us to represent the algorithm of the boundary functions method for constructing an asymptotic solution of initial-value singularly perturbed problems in the critical case more clearly than in [4]. Note that the projector approach has been used in [10] for constructing the zero-order asymptotic solution for a singularly perturbed linear-quadratic control problem in the critical case. We introduce orthogonal projectors of the space X onto kerA(t) and kerA(t)0 Based on these projectors, the algorithm of constructing the zero-order asymptotic approximation of a solution of problem (1)-(2).

Problem Decomposition

Space Decomposition

Zero-Order Asymptotic Solution

Higher-Order Asymptotic Solutions

Illustrative Example

Conclusions