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For switched linear systems with time delay in detection of switching signal, the authors investigate its L2 gain control synthesis problem via off-line-type switched state feedback by average dwell-time approach, that is identifying an average dwell time to guarantee that switched system is exponentially stable with an L2 gain less than chi. The authors first establish an estimation on the state transition matrix of switched systems. Then, based on this concise estimation and some famous mathematical inequalities such as Minkowski's integral inequality and Cauchy-Schwartz inequality, the authors prove that there exists a switched state-feedback controller such that switched system has an L2 gain less than chi for any switching signal with average dwell time greater than a given constant. Finally, an example is given to illustrate our results. Our results complement and generalise those results given in the works of Xie and Wang, Ji et al. and Xie and Chen.