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 one-dimensional 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.

This article addresses the modeling and control of heterogeneous populations of thermostatically controlled loads (TCLs) operated by model predictive control (MPC) schemes at the level of each TCL. It is shown that the dynamics of such TCLs populations can be described by a pair of Fokker-Planck equations coupled via the actions in the domain. The technique of input-output feedback linearization is used in the design of aggregate power control, which leads to a nonlinear system in a closed loop. Well-posedness analysis is carried out to validate the developed control scheme. The closed-loop stability of the system is assessed by rigorous analysis. A simulation study is contacted, and the obtained results confirm the validity and effectiveness of the proposed approach.

The aim of this paper is to introduce a weak maximum principle--based approach for studying the input-to-state stability (ISS) with respect to boundary disturbances and states in certain classes for a class of one-dimensional nonlinear parabolic partial differential equations (PDEs) with nonlinear boundary conditions. To tackle the difficulties in ISS analysis due to, in particular, the nonlinear terms on the boundary, we establish first several maximum estimates for the solutions of linear parabolic PDEs with different nonlinear boundary conditions by means of the weak maximum principle. Then, using the technique of splitting and combining maximum estimates for the solutions of linear parabolic PDEs and the Lyapunov method, we establish ISS estimates for nonlinear parabolic PDEs with nonlinear boundary conditions. Two examples of specific parabolic equations with nonlinear boundary conditions are provided to illustrate the developed approach.