Abstract—In Part I of this paper, we introduced the notion of L∞-vanishment and established several kinds of characterizations of this notion. Based on this notion, L∞ low gain feedback was reconsidered and a systematic approach to L∞ low gain feedback design was proposed. In this second part of the paper, we consider the parallel notion of L2-vanishment. We establish several characterizations of the L2-vanishment property, based on which a new design approach referred to as the L2 low gain feedback approach is developed. Just as the L∞ low gain design is useful in the control of linear systems subject to actuator magnitude saturation, the L2 low gain feedback is instrumental in the control of systems with control energy constraints. As applications of L2 low gain feedback, the problem of semi-global stabilization of linear systems with energy constraints and the problem of linear time-delay system with energy constraints are solved in this paper. The notion of L2-vanishment and the resulting L2 low gain feedback are also extended to nonlinear systems. As in the L∞ low gain feedback considered in Part I, a systematic approach is also developed for L2 low gain feedback design in this second part of the paper. Finally, an example involving a linearized model of the relative motion with respect to another in a circular orbit around the Earth is used to illustrate the effectiveness of the results developed in this paper.
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