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This paper deals with the issue of stochastic admissibility and robust extended dissipative control for uncertain delayed hybrid descriptor systems (DHDSs) under a unified framework. Belonging to a convex bounded domain, the parameter uncertainties are time-invariant and appear in the state matrix, input matrix and disturbance matrix. Firstly, a stochastic Lyapunov–Krasovskii functional is proposed to reflect the information of time-varying delays and Markovian jump modes, and singular value decomposition technique is utilized to derive novel criteria that ensure the stochastic admissibility and robust extended dissipativity of DHDSs. Then, under a unified framework, the desired robustly stabilizing state feedback controller is realized in terms of linear matrix inequalities. Finally, two examples including a numerical example and an oil catalytic cracking process are applied to corroborate the effectiveness of the derived synthesis scheme. [ABSTRACT FROM AUTHOR]