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In this paper, we introduce a new type of Lyapunov functions in a general framework particularly suitable for the analysis and control of systems with noise and uncertainty. These Lyapunov functions may depend on parameters possibly satisfying differential equations. The main differences with respect to classical Lyapunov functions and classical tools for the design of composite Lyapunov functions are discussed through examples. A design tool for the design of composite filtered Lyapunov functions is given, and examples show improvements over existing literature.