Motivated from linear-quadratic optimal control problems for differential-algebraic
equations (DAEs), we study the functional analytic properties of the operator associated
with the necessary optimality boundary value problem and show that it is associated
with a self-conjugate operator and a self-adjoint pair of matrix functions. We then study
general self-adjoint pairs of matrix valued functions and derive condensed forms under
orthogonal congruence transformations that preserve the self-adjointness. We analyze the
relationship between self-adjoint DAEs and Hamiltonian systems with symplectic flows.
We also show how to extract self-adjoint and Hamiltonian reduced systems from derivative
arrays.
