Decomposition of Hessenberg DAE systems to state space form

Abstract

An algorithm is given for symbolically decoupling the solutions to a linear, time dependent differential-algebraic equation Ez′ = A(t)z + ⨍(t), z(t)ϵR s , in Hessenberg form into state and algebraic components. The state variables are the solutions to an ordinary differential equation with initial conditions restricted to a subspace of R s , while the algebraic components are linear functions of the state variables involving derivatives of the coefficients and input functions up to order r − 1, where r is the index of the system. This decomposition provides closed form solutions to linear Hessenberg DAEs in terms of the fundamental solutions of the state variable system. The implications of the algorithm for computing consistent initial conditions, for certain singular optimal control problems, and for numerical solutions are briefly discussed.