This paper considers the finite-time \(H_{\infty }\) dynamic output feedback control for a class of one-sided Lipschitz nonlinear rectangular descriptor Markov jump systems (DMJSs). The differential matrix \(E\in \mathbb {R}^{m\times n}\) is not subject to any constraints, i.e., it includes two cases of \(m\ge n\) and \(m\le n\) . For making the closed-loop system as square DMJSs, the rectangular dynamic output feedback controller is proposed. Firstly, the sufficient conditions are given to guarantee that the augmented closed-loop systems are singular stochastic \(H_{\infty }\) finite-time bounded (SS \(H_{\infty }\) FTB) and have a unique solution simultaneously by adopting a mode-dependent Lyapunov functional and implicit function theorem. Then a novel and rigorous strict linear matrix inequality (LMI) sufficient condition for the existence of a rectangular dynamic output feedback controller is given based on the certain matrix decoupling techniques, and the controller is obtained. Ultimately, numerical examples are provided in order to substantiate the soundness of the results.
