Adjoint pairs of differential-algebraic equations and Hamiltonian systems

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 characterize the boundary conditions that lead to self-adjoint boundary value problems for the essentially underlying Hamiltonian systems.