Applied Numerical Mathematics 53 (2005) 131–148 www.elsevier.com/locate/apnum Adjoint pairs of differential-algebraic equations and Hamiltonian systems ✩ Katalin Balla a,∗ , Vu Hoang Linh b a Computer and Automation Research Institute, Hungarian Academy of Sciences, H-1518 Budapest P.O. Box 63, Hungary b Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, Hanoi, 334 Nguyen Trai Str., Vietnam Abstract We consider linear homogeneous differential-algebraic equations A(Dx) ′ + Bx = 0 and their adjoints −D ∗ (A ∗ x) ′ + B ∗ x = 0 with well-matched leading coefficients in parallel. Assuming that the equations are tractable with index less than or equal to 2, we give a criterion ensuring the inherent ordinary differential equations of the pair to be adjoint each to other. We describe the basis pairs in the invariant subspaces that yield adjoint pairs of essentially underlying ordinary differential equations. For a class of formally self-adjoint equations, we charac- terize the boundary conditions that lead to self-adjoint boundary value problems for the essentially underlying Hamiltonian systems.  2004 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Differential-algebraic equations; Adjoint pairs of differential-algebraic equations; Self-adjoint boundary value problems 1. Introduction Recently, differential-algebraic equations (DAEs) of the form A(Dx) ′ + Bx = f, (1) ✩ This work was supported by OTKA (Hung. National Sci. Foundation) Grants # T043276, T031807. * Corresponding author. E-mail addresses: balla@sztaki.hu (K. Balla), vhlinh@hn.vnn.vn (V.H. Linh). URL: http://www.sztaki.hu/~balla. 0168-9274/$30.00  2004 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2004.08.015