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The problems of robust stability and robust stabilization of linear state-space systems with parameter uncertainties have been extensively studied in the past decades [25, 209]. Many results on these topics have been proposed. Among the different approaches that deal with these problems, the methods based on the concepts of quadratic stability and quadratic stabilizability have become popular. An uncertain system is quadratically stable if there exists a fixed Lyapunov function to infer the stability of the uncertain system, while an uncertain system is quadratically stabilizable if there exists a feedback controller such that the closed-loop system is quadratically stable. Many results on quadratic stability and quadratic stabilizability have been reported in both the continuous and discrete contexts; see, e.g., [8, 57, 182, 210], and the references therein. However, in the context of uncertain singular systems, it has been shown that the problems of robust stability and robust stabilization are more complicated than those for state-space systems. Specifically, the robust stability problem for singular systems requires considering not only stability robustness, but also regularity and non-impulsiveness (for continuous singular systems) or causality (for discrete singular systems) [48, 49, 112, 177, 183], while the latter two are intrinsic properties of state-space systems. Similarly, the robust stabilization problem for uncertain singular systems must consider not only stabilization but also regularization and impulse elimination, while the latter two issues do not arise in the state-space case.
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