Weak input-to-state stability: characterizations and counterexamples

Abstract

We establish characterizations of weak input-to-state stability for abstract dynamical systems with inputs, which are similar to characterizations of uniform and of strong input-to-state stability established in a recent paper by A. Mironchenko and F. Wirth. We also answer, by means of suitable counterexamples, two open questions concerning weak input-to-state stability (and its relation to other common stability concepts) raised in the aforementioned paper.

Asymptotic gain results for attractors of semilinear systems

“…for all t, h ≥ 0, x ∈ X and v ∈ V. It should also be noticed that semiprocess families are closely related to the (forward-complete) dynamical systems with inputs from [29], [32]. Indeed, every semiprocess family (S u ) u∈U with input space U ⊂ L ∞ (R + 0 , U ) and translation T given by (2.2) determines a dynamical system (X, U, ϕ) with inputs in the sense of [29], [32] via…”

Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances

“…inputs from L 2 ([0, ∞), R k ) iff it is uniformly globally stable and of weak asymptotic gain. See [45] for other characterizations of weak input-to-state stability. In this context, the weak asymptotic gain property by definition means the following: there is a function γ ∈ K ∪ {0} such that for every ε > 0 and everyx 0 ∈X and…”

“…See [33], [7], [34], [35] and, respectively, [40], [21], [36] among many others. And finally, a multitude of papers have recently been published on fundamental characterizations and criteria for input-to-state stability [29], [30], [17], [15], [18], [19] and for versions of input-to-state stability like local [28], integral [15], strong [37], weak [45] and input-to-state practical stability [32], for instance. A nice survey of the state of the art of input-to-state stability for infinite-dimensional systems can be found in [36] -along with an extensive list of references on the topic.…”

Small gain theorems for networks of heterogeneous systems

“…Techniques developed within infinite-dimensional ISS theory include characterizations of ISS and ISS-like properties in terms of weaker stability concepts [26,23], [9,32], constructions of ISS Lyapunov functions for PDEs with distributed and boundary controls [21,31,4,25,37,42], non-coercive ISS Lyapunov functions [26,8], efficient methods for study of boundary control systems [41,9,10,17,20], etc.…”

“…Motivated by the notion of strong input-to-state stability (sISS), introduced in [26] and studied in [26], [28], in [32,33] the concept of weak input-to-state stability (wISS) has been introduced and investigated, in particular, in the context of robust stabilization of port-Hamiltonian systems, see [33]. The system is called weak ISS, if it is uniformly globally stable and has an asymptotic gain property.…”