Boundary conditions and their transfer for differential-algebraic equations of index 1 (Q1921226)

The purpose of this paper is to describe the linear subspace of the differential-algebraic equation (1) \(Ay' + By = 0\) defined by the condition (2) \(C^T y(x_0) = 0\), where \(A,B \in C^1 (I,L (\mathbb{R}^n))\), \(C \in L(\mathbb{R}^m, \mathbb{R}^n)\), \(I \subset \mathbb{R}\) is a closed interval, \(x_0 \in I\) and equation (1) is index 1-tractable with \(\dim \text{Im} A = r < n\). To obtain the adjoint or dual of (1), i.e., (3) \((A^T \eta)' - B^T \eta = 0\), the author applies the reduction method. Equation (3) transfers the condition (2) along the interval I. Above the theoretical interest, this study is important for elaborating numerical methods to solve to boundary value problems (because a boundary value problem for a differential-algebraic equation may be reduced to that of a regular ordinary differential equation).