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

Abstract

This paper discusses the boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and in the presence of a time-varying state-delay. The proposed control design strategy is based on a finite-dimensional truncated model obtained via a spectral decomposition. By an adequate selection of the number of modes of the original infinitedimensional system, we show that the design performed on the finite-dimensional truncated model achieves the exponential stabilization of the original infinite-dimensional system. In the presence of distributed disturbances, we show that the closedloop system is exponentially input-to-state stable with fading memory. (Hugo Lhachemi), robert.shorten@ucd.ie (Robert Shorten).diagonal infinite-dimensional systems in [13,16] for constant input delays and then in [14] for fast time-varying input delays. A second research direction deals with the boundary feedback stabilization of PDEs in the presence of a state-delay. Motivated by the success of Linear Matrix Inequalities (LMI)-based approaches for the study of delayed finite-dimensional systems [3], LMI conditions were investigated in [4,26] for the stability analysis of PDEs in the presence of a state-delay. For the boundary control design of state-delayed PDEs, backsteppingbased methods were reported in [6,7,8,9].

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).…”

“…Remark 3. Examples of systems covered by Assumptions 1-3 and thus for which the proposed control strategy applies include reaction-diffusion equations [22,30], linear Kuramoto-Sivashinsky equation [11], and certain damped flexible string or beam models [9, Ex. 2.23, p. 91] [23].…”

Proportional Integral Regulation Control of a One-Dimensional Semilinear Wave Equation

“…, where h > 0 is a state-delay. Possible approaches include the use of either Lyapunov-Krasovskii functionals [25] small gain arguments [30].…”

Robustness of constant-delay predictor feedback for in-domain stabilization of reaction–diffusion PDEs with time- and spatially-varying input delays

“…Stabilization of open-loop unstable partial differential equations (PDEs) in the presence of delays has attracted much attention in the recent years. A first class of problems deals with the feedback stabilization of PDEs in the presence of a state-delay [12,[15][16][17][18]26,25,41]. In this paper, we are concerned with a second class of problem, namely: the feedback stabilization of PDEs in the presence of a delay in the control input [14,[21][22][23]28,27,24,[31][32][33][34][35].…”