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(eng): This paper presents a momentous aspect that should be considered in the stability analysis and synthesis of switched systems by investigating an N-step ahead Lyapunov function (LF) approach. By considering these traits, the proposed incorrect theorems in the aforementioned paper are revised to new ones. An illustrative numerical example manifests the effectiveness of our proposed approach. Unlike the assertion in the aforementioned paper when the monotonic requirement of LF is relaxed, the switching is not necessarily gentler. Also, our results indicate that our approach guarantees lower average dwell time.