Submitted

@unpublished{AhmeBach25pp,

title = {Lyapunov Characterization for ISS of Impulsive Switched Systems},

author = {Saeed Ahmed and Patrick Bachmann and Stephan Trenn},

url = {https://stephantrenn.net/wp-content/uploads/2025/12/Preprint-ABT251201.pdf, Preprint},

year  = {2025},

date = {2025-12-01},

urldate = {2025-12-01},

abstract = {In this study, we investigate input-to-state stability (ISS) of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is not necessarily decreasing along trajectories, which we call generalized, and another one that is decreasing. Our research proves that the existence of either of these ISS-Lyapunov functions is a necessary and sufficient condition for ISS. We also present a strictification technique for constructing a decreasing ISS-Lyapunov function from a large class of generalized ones, which is useful for its own sake. Our findings also have added value to previous research that only studied sufficient conditions for ISS, as our results apply to a broader class of systems. This is because we impose less restrictive dwell and leave time constraints on the switching signal and our ISS-Lyapunov functions are time-varying with general nonlinear conditions imposed on them. Moreover, we provide a method to guarantee ISS of a particular class of impulsive switched systems when the switching signal is unknown.},

note = {conditionally accepted in IEEE Transactions on Automatic Control as regular paper},

keywords = {},

pubstate = {published},

tppubtype = {unpublished}

}

@unpublished{TrenHaid25pp,

title = {On the equivalence of Lyapunov exponents for switched DAEs and corresponding singular perturbed system},

author = {Stephan Trenn and Ihab Haidar and Paolo Mason and Mario Sigalotti},

url = {https://stephantrenn.net/wp-content/uploads/2025/12/Preprint-THMS251114.pdf, Preprint},

year  = {2025},

date = {2025-11-14},

urldate = {2025-11-14},

abstract = {We are interested in the relationship between the stability properties of a singularly perturbed switched system and those of the corresponding limiting switched system. In some recent work, it has been shown that for a large class of singularly perturbed systems the worst case exponential growth rate for sufficiently small perturbation parameters is lower bounded by the growth rate of the limiting switched system. However, examples show that there could be a positive gap between these bounds. Here we want to investigate this gap further by first observing that the limiting switching system is in fact a switched differential-algebraic equation (switched DAE) for which numerous stability results are already available. Based on the underlying geometric structure of the switched DAE we introduce the concept of structurally aligned singular perturbations and show for the commuting case that indeed there is no gap. We also provide an example which has a commuting limiting switched DAE, but for which the singular perturbations are not structurally aligned and there is indeed a gap between the growth bounds.},

note = {submitted},

keywords = {},

pubstate = {published},

tppubtype = {unpublished}

}

@unpublished{CimpSter25pp,

title = {ISS for PWA systems},

author = {Radu Cimpean and Alef Sterk and Stephan Trenn},

url = {https://stephantrenn.net/wp-content/uploads/2025/12/Preprint-CST251114.pdf, Preprint},

year  = {2025},

date = {2025-11-14},

urldate = {2025-11-14},

abstract = {We offer a way to construct an ISS type bound for a PWA system. Starting from an ISS Lyapunov function in each region of the state space (called regional ISS Lyapunov function), and under suitable assumptions, we patch these functions and obtain initially a practical ISS type bound. After that, depending on the input size, we have to consider a distinction between large and small inputs. The large inputs dominate the dynamics and may induce region transitions, while the small inputs are sufficiently attenuated to direct and keep the trajectory within the region containing the equilibrium. The threshold between these regimes plays a key role in determining the point at which the system decays towards the equilibrium. By unifying the analysis across these input regimes, we provide a complete ISS-type bound valid for all admissible inputs. Consequently, the global ISS property is ensured, and further refined analysis can be applied.},

note = {submitted},

keywords = {},

pubstate = {published},

tppubtype = {unpublished}

}