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There is limited formal mathematical analysis of one type of games - dynamic sequential games with large, or even infinitely large, planning horizons, from the point view of system controls. In this paper, we consider a class of noncooperative dynamic linear quadratic sequential games (LQSGs). For LQSGs with finite planning horizons, we provide state feedback Nash strategies, and their existence and uniqueness within the class of state feedback strategies are proved. When the planning horizon approaches infinity, we prove that the feedback systems with the state feedback Nash strategies are uniformly asymptotically stable, given that the associated coupled Riccati equations have a positive definite solution. Finally we show that at least one positive definite solution for the coupled Riccati equations of a scalar LQSG exists.