This paper investigates the extended stabilization control problem for continuous-time descriptor systems (also refereed to as singular systems, implicit systems or generalized state-space systems) where unstable and nonproper weighting filters are used. In such nonstandard problem, the desired controller, called the extended stabilizing controller, has to satisfy two conditions simultaneously: (i) the closed-loop is admissible; (ii) only the internal stability of a part of the closed-loop is required. In terms of two generalized Sylvester-type equations, necessary and sufficient conditions for the existence of an observer-based extended stabilizing controller are given. Moreover, a parametrization of the class of extended stabilizing controllers is formulated.