The index of linear differential algebraic equations with properly stated leading terms

Abstract

In a linear differential algebraic equation with properly stated leading term, the involved derivatives of the unknown function are figured out by an additional matrix coefficient being in some sense well matched with the leading coefficient matrix. Supposed all matrix coefficients are continuous, for these equations, the notion of regularity with index $\mu$ is introduced via a certain matrix sequence built up from the coefficient matrices and then proved to be invariant under regular transformations. The Kronecker index, the global index, and the tractability index for standard form differential algebraic equations are covered as well. Moreover, inherent regular ordinary differential equations that govern the dynamics are described in detail and it is shown that transformations applied to them are closely related to refactorizations of the leading term in the original equation.