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

Abstract

This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in L 2 -norm for Burgers' equation have been established using this method. Moreover, as an application of De Giorgi iteration, ISS in L ∞ -norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-D linear unstable reaction-diffusion equation have also been established. It is the first time that the method of De Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear partial differential equations (PDEs).

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

“…Based on (35)(36)(37)(38), we deduce from (34), Young's inequality, estimates (2), and the fact ‖ (0)…”

“…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

“…where c > 0 can be an arbitrary constant. Then using the techniques in this paper and arguing as [31,Section V], one may establish the maximum estimate for the solution of the above system, and hence for (6).…”

“…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:…”

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.…”