A compendium of comparison function results

Abstract

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].

The converging-input converging-state property for Lur’e systems

“…By [39,Proposition 7], it follows that a discontinuous KL-function can be bounded from above by a continuous one. For more details on comparison functions the reader is referred to [22].…”

“…Then it is not difficult to show that there exists f ∈ F such that 22) and it is clear that condition (A) does not hold for ξ = ξ 0 . We claim that, for…”

Input-to-state stability of Lur’e systems

“…It is known that a discontinuous KL-function can be bounded from above by a continuous KL-function, see [24,Proposition 7]. For more details on comparison functions, we refer the reader to [15].…”

On the Relation Between Turnpike Properties for Finite and Infinite Horizon Optimal Control Problems

“…Here, the infinite horizon definition was inspired by the usual notion of asymptotic stability (in its formulation via KL-functions which has become standard in nonlinear control, see [9]), while the finite horizon definition can be seen as an extension of the exponential turnpike property established in [3] under a strict dissipativity condition. Like in the previous sections, we will then be able to show that these two conditions are equivalent under suitable regularity conditions on the optimal control problem (2.1).…”