The use of comparison functions has become standard in systems and control theory, particularly for the purposes of studying stability properties. The use of these functions typically allows elegant and succinct statements of stability properties such as asymptotic stability and input-to-state stability and its several variants. Furthermore, over the last 20 years several inequalities involving these comparison functions have been developed that simplify their manipulation in the service of proving more significant results. Many of these inequalities have appeared in the body of proofs or in appendices of various papers. Our goal herein is to collect these inequalities in one place. Keywords Comparison functions · Stability theory · Nonlinear systems HistoryJose Massera appears to have been the first scholar to introduce comparison functions to the study of stability theory in 1956 [26]. In particular, to elegantly capture the notion of (local) positive definiteness he relied on a function a : R ≥0 → R ≥0 with "a(r ) being continuous and increasing when r > 0, a(0) = 0."To further describe a function having an infinitely small upper bound, he then used a function b : R ≥0 → R ≥0 "having the same properties of a(r )". 1 1 It is worth remarking that in his 1949 manuscript studying similar stability problems, Massera [25] does not use comparison functions, but instead uses the more classical ε−δ formulations, indicating their development sometime between 1949 [25] and 1956 [26].

Abstract-The recent rapid uptake of residential solar photovoltaic (PV) installations provides many challenges for electricity distribution networks designed for one-way power flow from the distribution company to the residential customer. For gridconnected installations, intermittent generation as well as large amounts of generation during low load periods can lead to a degradation of power quality and even outages due to overvoltage conditions. In this paper we present four control methodologies to mitigate these difficulties using small-scale distributed battery storage. These four approaches represent three different control architectures: centralized, decentralized, and distributed control. These approaches are validated and compared using data on load and generation profiles from customers in an Australian electricity distribution network.

We derive the capacity of the binary multi-way relay channel, in which multiple users exchange messages at a common rate through a relay. The capacity is achieved using a novel functional-decode-forward coding strategy. In the functional-decode-forward coding strategy, the relay decodes functions of the users' messages without needing to decode individual messages. The functions to be decoded by the relay are defined such that when the relay broadcasts the functions back to the users, every user is able to decode the messages of all other users.

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We consider stability with respect to two measures of a difference inclusion, i.e., of a discrete-time dynamical system with the push-forward map being set-valued. We demonstrate that robust stability is equivalent to the existence of a smooth Lyapunov function and that, in fact, a continuous Lyapunov function implies robust stability. We also present a sufficient condition for robust stability that is independent of a Lyapunov function. Toward this end, we develop several new results on the behavior of solutions of difference inclusions. In addition, we provide a novel result for generating a smooth function from one that is merely upper semicontinuous.