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 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.

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.

Due to the compact layout, manufacturing tolerance, modeling errors, and environmental changes, microelectromechanical systems (MEMSs) are subjected to parasitics and parameter variations. In order to better guarantee their stability and a certain level of performance, one must take into account these factors in the design of MEMS control systems. This work presents two robust control laws for a parallel-plate electrostatic microactuator in the presence of uncertainties. The dynamical model of the system, including parallel and serial parasitics, is firstly established and two control schemes, both based on input-to-state stabilization and robust backstepping, are proposed. The stability and the performance of the system using these control schemes are demonstrated through both stability analysis and numerical simulation.