This paper considers a non-standard J-spectral factorization (NS-J-SFP)
of a rational spectral matrix, which is related to the non-standard
H_\infty control problem for a descriptor system. We derive necessary
and sufficient conditions for the existence of a semi-stabilizing
solution of a generalized algebraic Riccati equation (GARE). We then
develop the solvability condition for the NS-J-SFP by adapting the zero
compensation technique of Copeland and Safonov [7] and Xin and Kimura
[26] to the descriptor system. Thus we can conclude that the NS-J-SFP
is solvable if and only if the GARE has a semi-stabilizing solution.
A numerical result is also included.