ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs

Abstract

Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-tostate stability (ISS) properties and finite gains cannot be established with respect to disturbances at the boundary. This intuition has been reinforced by many unsuccessful attempts, as well as by the success in establishing ISS only with respect to the derivative of the disturbance. Contrary to this intuition, we establish such a result for parabolic PDEs. Our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time-varying subsets of the state space. The obtained results are used for the comparison of the gain coefficients of transport PDEs with respect to inlet disturbances and for the establishment of the ISS property with respect to control actuator errors for parabolic systems under boundary feedback control.

A De Giorgi Iteration-Based Approach for the Establishment of ISS Properties for Burgers’ Equation With Boundary and In-domain Disturbances

“…Remark 4: As pointed out in [16], the assumptions on the continuity of f and d are required for assessing the existence of a classical solution of the considered system. However, they are only sufficient conditions and can be weakened if solutions in a weak sense are considered.…”

“…To resolve this concern while not invoking unbounded operators in the analysis, it is proposed in [15], [16], [17] to derive the ISS property directly from the estimates of the solution to the considered PDEs using the method of spectral decomposition and finite-difference. ISS in L 2 -norm and in weighted L ∞ -norm for PDEs with a Sturm-Liouville operator is established by applied this method in [15], guchuan.zhu@polymtl.ca This paper has been accepted for publication by IEEE TAC, and is available at http://dx.doi.org/10.1109/TAC.2018.2880160 [16], [17]. However, spectral decomposition and finite-difference schemes may involve heavy computations for nonlinear PDEs or problems on multidimensional spatial domains.…”

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.…”

A Maximum Principle-based Approach for Input-to-State Stability Analysis of Parabolic Equations with Boundary Disturbances

“…To tackle this issue, different solutions have been developed recently [7,10,11,12,13,14,15,26,31,32,33,34]. The approach considered in the present work is in line with the methods aimed at establishing the ISS properties from the a priori estimates of the solutions to the PDEs, including:…”