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In this manuscript, we consider the finite-time <inline-formula> <math display="inline"> <semantics> <msub> <mi>H</mi> <mo>∞</mo> </msub> </semantics> </math> </inline-formula> control for nonlinear systems with time-varying delay. With the assistance of a novel Lyapunov-Krasovskii functional which includes some integral terms, a matrix-based on quadratic convex approach, combined with Wirtinger inequalities and some useful integral inequalities, a sufficient condition of finite-time boundedness is established. A novel feature presents in this paper is that the restriction which is necessary for the upper bound derivative is not restricted to less than 1. Further a <inline-formula> <math display="inline"> <semantics> <msub> <mi>H</mi> <mo>∞</mo> </msub> </semantics> </math> </inline-formula> controller is designed via memoryless state feedback control and a new sufficient conditions for the existence of finite-time <inline-formula> <math display="inline"> <semantics> <msub> <mi>H</mi> <mo>∞</mo> </msub> </semantics> </math> </inline-formula> state feedback for the system are given in terms of linear matrix inequalities (LMIs). At the end, some numerical examples with simulations are given to illustrate the effectiveness of the obtained result.