ISS implies iISS even for switched and time-varying systems (if you are careful enough) ✩ Hernan Haimovich a,∗ , José Luis Mancilla-Aguilar b a International French-Argentine Center for Information and Systems Science (CIFASIS), CONICET-UNR, Ocampo y Esmeralda, 2000 Rosario, Argentina b Departamento de Matemática, Instituto Tecnológico de Buenos Aires, Av. Eduardo Madero 399, Buenos Aires, Argentina Keywords: Nonlinear systems Converse theorems Time-varying systems Switched systems Input-to-state stability abstract For time-invariant systems, the property of input-to-state stability (ISS) is known to be strictly stronger than integral-ISS (iISS). Known proofs of the fact that ISS implies iISS employ Lyapunov characteri- zations of both properties. For time-varying and switched systems, such Lyapunov characterizations may not exist, and hence establishing the exact relationship between ISS and iISS remained an open problem, until now. In this paper, we solve this problem by providing a direct proof, i.e. without requiring Lyapunov characterizations, of the fact that ISS implies iISS, in a very general time-varying and switched-system context. In addition, we show how to construct suitable iISS gains based on the comparison functions that characterize the ISS property, and on bounds on the function f defining the system dynamics. When particularized to time-invariant systems, our assumptions are even weaker than existing ones. Another contribution is to show that for time-varying systems, local Lipschitz continuity of f in all variables is not sufficient to guarantee that ISS implies iISS. We illustrate application of our results on an example that does not admit an iISS-Lyapunov function. 1. Introduction Both input-to-state stability (ISS) and integral-input-to-state stability (iISS) can be considered nonlinear-system extensions of the type of stability that a linear time-invariant system with inputs is known to have. The norm of the state of a system that is either ISS or iISS can be bounded by the sum of a term depending only on the initial state norm and decaying asymptotically to zero, and a term depending only on the input. Loosely speaking, for ISS the input-dependent term depends on a bound on the maximum input amplitude whereas for iISS, the dependence is on the input energy. The concepts of ISS and iISS, originally introduced for time- invariant continuous-time systems in, respectively, Sontag (1989, 1998), were subsequently extended and studied for other classes of systems: time-varying systems (Edwards, Lin, & Wang, 2000), discrete-time systems (Jiang & Wang, 2001), switched systems (Haimovich & Mancilla-Aguilar, 2018; Mancilla-Aguilar & García, 2001), impulsive systems (Hespanha, Liberzon, & Teel, 2008), ✩ Work partially supported by ANPCyT under grant PICT 2014-2599, Ar- gentina. The material in this paper was not presented at any conference.This paper was recommended for publication in revised form by Associate Editor Debasish Chatterjee under the direction of Editor Daniel Liberzon. ∗ Corresponding author. hybrid systems (Cai & Teel, 2009; Noroozi, Khayatian, & Geisel- hart, 2017) and infinite dimensional systems (Dashkovskiy & Mironchenko, 2013; Mironchenko & Wirth, 2018). For time-invariant systems, ISS was shown to be equivalent to the existence of a dissipation-form ISS-Lyapunov function (Son- tag & Wang, 1995), and analogously for iISS (Angeli, Sontag, & Wang, 2000a). Based on these Lyapunov characterizations of ISS and iISS, it is easy to see that if a system is ISS, then it also is iISS (Angeli et al., 2000a). In other words, for time-invariant systems, it is known that ISS implies iISS. The only requirement for the latter implication to hold is that the function f defining the system dynamics ˙ x = f ti (x, u) be locally Lipschitz. Loosely speaking, classes of systems where ISS is equivalent to the ex- istence of a dissipation-form ISS-Lyapunov function make the implication ISS ⇒ iISS hold. For such classes of systems, the latter implication can be established almost identically as for continuous-time time-invariant systems. Some of these classes are, for example, time-invariant switched systems under arbitrary switching (Mancilla-Aguilar & García, 2001) and time-invariant hybrid systems (Cai & Teel, 2009; Noroozi et al., 2017). Although many other characterizations of ISS (Liberzon & Shim, 2015; Sontag & Wang, 1995, 1996) and iISS (Angeli et al., 2000a; Angeli, Sontag, & Wang, 2000b; Haimovich & Mancilla- Aguilar, 2018) exist, until now the only known way of proving that ISS implies iISS was based on a dissipation-form ISS-Lyapunov function as mentioned above. Therefore, classes of systems for which ISS is not necessarily equivalent to the existence of a E-mail addresses: haimovich@cifasis-conicet.gov.ar (H. Haimovich), jmancill@itba.edu.ar (J.L. Mancilla-Aguilar).