Infinite-Dimensional Input-to-State Stability and Orlicz Spaces

Abstract

Abstract. In this work, the relation between input-to-state stability and integral input-to-5 state stability is studied for linear infinite-dimensional systems with an unbounded control operator. 6Although a special focus is laid on the case L ∞ , general function spaces are considered for the inputs. 7We show that integral input-to-state stability can be characterized in terms of input-to-state stability 8 with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated 9 in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability notions 10 with respect to L ∞ are equivalent. 11Key words. Input-to-state stability, integral input-to-state stability, C 0 -semigroup, admissibil-12 ity, Orlicz spaces 13 AMS subject classifications. 93D20, 93C05, 93C20, 37C75 14

An operator theoretic approach to infinite‐dimensional control systems

“…( [31]) Assume that the port-Hamiltonian system (4) is a well-posed boundary control and observation system and that the corresponding operator A is exponentially stable. ( [31]) Assume that the port-Hamiltonian system (4) is a well-posed boundary control and observation system and that the corresponding operator A is exponentially stable.…”

“…For well-posed boundary control and observation systems input-to-state stability is equivalent to exponential stability of the corresponding semigroup. [31] Here we give an input-to state stability estimate for port-Hamiltonian systems.…”

“…Theorem 10.8. ( [31]) Assume that the port -Hamiltonian system (4) is a well-posed boundary control and observation system and that the corresponding operator A is exponentially stable. Then there exists constants M ≥ 1, > 0 and c > 0 such that for every initial condition x 0 ∈ X and every input function u ∈ L ∞ ((0, ∞); C m ) the corresponding mild solution x satisfies for every t ≥ 0…”

Exponential input-to-state stabilization of a class of diagonal boundary control systems with delay boundary control

“…This property also plays a key role in the establishment of small gain conditions for the stability of interconnected systems [17]. Although the study of ISS properties of finite-dimensional systems has been intensively studied during the last three decades, its extension to infinite-dimensional systems, and in particular with respect to boundary disturbances, is more recent [4,12,13,15,16,17,21,24,26,28,29,37,38]. Moreover, most of these results deal with the establishment of ISS properties for open-loop stable distributed parameter systems.…”

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.…”