An unified approach to linear differential algebraic equations and their adjoint equations

Abstract

Instead of a single matrix occuring in the standard setting the leading term of the linear differential equation is composed of a pair of well matched matrices. This problem formulation enables an unified treatment of the original equation and its adjoint one. An index notion is proposed for the equations and solvability statements for the lower index case are proven. The original equation and its adjoint are shown to have the same index. Their fundamental solution matrices satisfy a relation that generalizes the classical Lagrange identity. The coefficients are assumed to be just continuous and only certain subspaces have to be continuously differentiable additionally. Solution representations are given that base on the solutions of certain inherent regular ODEs which are uniquely determined by the problem data.