On Some Properties of Linear Mapping Induced by Linear Descriptor Differential Equation

TL;DR

This paper studies the properties of a linear mapping induced by linear descriptor differential equations, proving its closedness, constructing its adjoint, and establishing conditions for solutions and the closure of its range.

Contribution

It introduces and analyzes the linear mapping associated with descriptor differential equations, including its adjoint, and provides criteria for solutions and the closure of its range.

Findings

D is a closed dense mapping for any m×n matrix F.

Constructed the adjoint mapping D* and described its domain.

Provided necessary and sufficient conditions for the existence of generalized solutions.

Abstract

In this paper we introduce linear mapping D from WnF\subset Ln into Lm\times Rm, induced by linear differential equation d/dt Fx(t)-C(t)x(t)=f(t),Fx(t_0)=f_0. We prove that D is closed dense defined mapping for any m\times n-matrix F. Also adjoint mapping D* is constructed and its domain WmF is described. Some kind of so-called "integration by parts" formula for vectors from WnF, WmF is suggested. We obtain a necessary and sufficient condition for existence of generalized solution of equation Dx=(f,f_0). Also we find a sufficient criterion for closureness of the R(D) in Lm\times Rm which is formulated in terms of transparent conditions for blocks of matrix C(t). Some examples are supplied to illustrate obtained results.