This paper addresses a new framework of control design, which allows for parameter-dependent Lyapunov functions for continuous-time systems. The control design is performed in an extended space rather than the space where the problem is originally given. Inverse use of the elimination lemma can explore this framework. This paper focuses on the H₂/H_∞ problem in multiobjective control design, robust H₂ performance against polytopic uncertainties, and gain-scheduled control for linear parameter-varying systems. However, more other control problems could be handled with this framework. Through an illustrated example of a two-mass, one-spring, and one-damper mechanical system, the applicability of the proposed method is demonstrated.