Stability analysis of reaction–diffusion PDEs coupled at the boundaries with an ODE

Saturated Boundary Stabilization of Partial Differential Equations Using Control-Lyapunov Functions

“…Some simulations have illustrated our results and design methods. More recent works follow the present chapter as the control of reaction-diffusion equation coupled with ordinary differential equations (see 35 ), or control of such partial differential equation by means of delayed control (see 42 ) to cite just a few. As far as hyperbolic system are considered, nonlinear controllers could be also designed as done in.…”

“…Here ζ(t) is defined by ζ(t) = i≥N +1 Φ i (0)w i (t) while, following, 41 we introduced the scaled error of observation ẽn (t) = √ λ n e n (t) with e n given by (35). The definitions of C 0 and C 1 are replaced by…”

Anti-Windup Design for a Reaction-Diffusion Equation

“…Inequality (12) corresponds to an input-output stability (IOS) bound for the closed-loop system. The main contribution of this paper is to design an anti-windup system K a in order to further minimize the effect of the gain ρ a for d ̸ = 0 and further maximize the region of attraction R a for d = 0.…”

“…The minimization problem of the effect of the external perturbations on system (3) in closed loop with (8) and ( 9) boils down to designing the anti-windup with minimal ρ a in (12). Due to (21), such minimization can be achieved by solving the following optimization problem:…”

“…In this work, we focus on the output feedback stabilization of a specific type of partial differential equation called the reaction-diffusion equation. The global stability of a reactiondiffusion equation has been investigated in previous papers [7]- [9], [12], [19] and the extension to local stability in the case of saturated actuator was studied in [13]. We will use the regional, LMI-based anti-windup scheme presented in [21], [22] to achieve local exponential input-output stability for a class of distributed parameter systems.…”