Input-to-State Stability for PDEs

Abstract

Communications and Control Engineering is a high-level academic monograph series publishing research in control and systems theory, control engineering and communications. It has worldwide distribution to engineers, researchers, educators (several of the titles in this series find use as advanced textbooks although that is not their primary purpose), and libraries.The series reflects the major technological and mathematical advances that have a great impact in the fields of communication and control. The range of areas to which control and systems theory is applied is broadening rapidly with particular growth being noticeable in the fields of finance and biologically-inspired control. Books in this series generally pull together many related research threads in more mature areas of the subject than the highly-specialised volumes of Lecture Notes in Control and Information Sciences. This series's mathematical and control-theoretic emphasis is complemented by Advances in Industrial Control which provides a much more applied, engineering-oriented outlook.

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

“…Combining (22)(23)(24), we deduce the existence of constants Step 4: derivation of the claimed estimate (17). To conclude, it remains to show that sup ∈[0,max( ,2 )] ‖ ( )‖ can be replaced by sup (25).…”

“…boundary disturbances comparing to distributed ones seems to be a global trend for infinite-dimensional systems [27]. In this paper, under the assumption of a sector condition on the eigenvalues corresponding to the modes which are not captured by the truncated model used for the design of the predictor feedback, we show that the resulting infinitedimensional closed-loop system is exponentially ISS with fading memory [17] of the boundary disturbances for small variations of the time-varying delay around its nominal value. The adopted approach relies first on the extension of a small gain argument reported in [14] in order to establish the ISS property of the closed-loop truncated model, and then on the method reported in [24] for the establishment of ISS estimates with respect to boundary disturbances for diagonal infinite-dimensional systems.…”

“…The main motivation in achieving an input-to-state stabilization of the closed-loop system relies in the fact that the Input-to-State Stability (ISS) property, originally introduced by Sontag in [36], is one of the main tools for assessing the robustness of a system with respect to boundary disturbances. 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].…”

Boundary feedback stabilization of a reaction–diffusion equation with Robin boundary conditions and state-delay

“…In particular, the proposed approach allows either one single command input (located at one of the two boundaries of the domain) or two command inputs. Second, we show that the resulting closedloop system is, in the presence of distributed boundary disturbances, exponentially Input-to-State Stable (ISS) with fading memory [11]. The concept of ISS, originally introduced by Sontag in [27], plays an important role in the robustness assessment of dynamical systems, as well as the stability of interconnected systems [11].…”

“…Second, we show that the resulting closedloop system is, in the presence of distributed boundary disturbances, exponentially Input-to-State Stable (ISS) with fading memory [11]. The concept of ISS, originally introduced by Sontag in [27], plays an important role in the robustness assessment of dynamical systems, as well as the stability of interconnected systems [11]. The extension of this notion to infinite-dimensional systems raises many challenges [19,20] and is the topic of very active research activities.…”