We present a numerical algorithm, based on using orthogonal matrices, and sufficient existence conditions for an LQ control problem of descriptor systems, which generalise the existing results. The associated Hamiltonian pencil may be singular and may have infinite generalised eigenvalues (IGEs) of multiplicity 1. Besides on x(0(+)), we present optimality conditions on x(0(-)). We present sufficient conditions for stability of the optimal closed-loop descriptor system, as well as a stabilisation algorithm.