Input-to-State Stability of a Clamped-Free Damped String in the Presence of Distributed and Boundary Disturbances

Abstract

This note establishes the Exponential Input-to-State Stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades, establishing ISS properties with respect to boundary disturbances remains challenging. One of the well-known methods for well-posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one. However, the resulting distributed disturbance involves time derivatives of the boundary perturbation. Thus, the subsequent ISS estimate depends on its amplitude, and may not be expressed in the strict form of ISS properties. To solve this problem, we show for a clamped-free damped string equation that the projection of the original system trajectories in an adequate Riesz basis can be used to establish the desired EISS property.

ISS Property with respect to boundary disturbances for a class of Riesz-spectral boundary control systems

“…In this Subsection 3.1, we develop a different approach for assessing such a result for classical solutions. The proposed approach generalizes the ideas developed in [19,23] consisting in the projection of the system trajectories over adequate Riesz bases. This approach relies on a novel spectral decomposition (see (9) and Remark 2) and can be used either to derive general (see proof of Theorem 1) or system specific (see Subsection 5.3) versions of the ISS estimates.…”

“…Among the examples of applications, one can find 1D parabolic PDEs [19] and a flexible damped string [23].…”

“…On one hand, this idea of projecting the system trajectories over a Hilbert basis was then investigated in [17] for the study of the asymptotic gains of a damped wave equation. On the other hand, this approach was further investigated in [23] for the study of the ISS properties of a clamped-free damped string. Speciffically, the system trajectories were projected over a Riesz basis [5] formed by eigenvectors of the disturbance free operator.…”

“…The ISS property of such analytic semigroups was investigated first in [14,15,16] for L ∞ boundary inputs. The problem was embedded into the extrapolation space H −1 while invoking admissibility conditions for returning to the original Hilbert space H. The approach adopted in this paper differs by generalizing the ideas developed first in [19] and then in [23]. Assuming boundary and distributed distur-bances of class C 2 and C 1 , respectively, the ISS property is established for classical solutions by taking advantage of the projection of the system trajectories over a Riesz basis formed by the eigenvectors of the disturbance free operator.…”

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.…”