Lyapunov-based boundary feedback design for parabolic PDEs

Abstract

This paper presents a methodology for the construction of simple Control Lyapunov Functionals (CLFs) for boundary controlled parabolic Partial Differential Equations (PDEs). The proposed methodology provides functionals that contain only simple (and not double or triple) integrals of the state. Moreover, the constructed CLF is "almost diagonal" in the sense that it contains only a finite number of cross-products of the (generalized) Fourier coefficients of the state. The methodology for the construction of a CLF is combined with a novel methodology for boundary feedback design in parabolic PDEs. The proposed feedback design methodology is Lyapunov-based and the feedback controller is an "integral" controller with internal dynamics. It is also shown that the obtained simple CLFs can provide nonlinear boundary feedback laws which achieve global exponential stabilization of semilinear parabolic PDEs with nonlinearities that satisfy a linear growth condition.

Stabilization of underactuated linear coupled reaction-diffusion PDEs via distributed or boundary actuation

“…Below, we apply a state transformation in order to place the control internally. Such a type of transformation, but for the scalar PDE, has been introduced in [21] leading to dynamic extension. We adapt this kind of transformation to the present vector case with one control.…”

“…We invoke first existence-uniqueness of solutions to the closed loop system (1), (51) with θ = 0 by easily adapting a result given in [21] for the scalar case to our vector case (proof of Theorem 2.2 therein). More precisely, for any given initial condition z 0 ∈ H 2 (0, L; R m ) satisfying γ 11 z 0 (0) + (1 − γ 11 ) z 0 (0) = γ 21 z 0 (L) + (1 − γ 21 ) z 0 (L) = 0 (implying by (38) that w 0 (•) := w(0, •) ∈ H 2 (0, L; R m ) satisfying γ 11 w 0 (0) + (1 − γ 11 ) w 0 (0) = γ 21 w 0 (L) + (1 − γ 21 ) w 0 (L) = 0) and input initial conditions u(0) = 0, there exists a unique solution w ∈ C 0 ([0, +∞) × [0, L]; R m ) ∩ C 1 ((0, +∞) × [0, L]; R m ) with w(t, •) ∈ C 2 ([0, L]; R m ) of the closed loop system (39), (69) implying also unique existence of z in the same function spaces due to (38).…”

“…We assume that the control placed on the right boundary of the first state is written as a sum of control components. Inspired by the recent dynamic extension approach in [21] for the scalar case, we adapt similar transformation to our vector case. The system is first mapped into a system where the control components and their time-derivatives are placed internally in the PDEs.…”

Global Output Feedback Stabilization of Semilinear Reaction-Diffusion PDEs

“…The extension of these approaches to the stabilization of semilinear reaction-diffusion PDEs remains challenging. Among the reported contributions, one can find the study of stability by means of strict Lyapunov functionals [19], observer design [11], [20], [29], and control using quasi-static deformations [3], state-feedback [8] or network control [27], [28].…”

Global finite-dimensional observer-based stabilization of a semilinear heat equation with large input delay

“…State-feedback control of some semilinear PDEs was studied in [33] using backstepping, in [15] using small-gain theorem and in [13] via control Lyapunov functions. Recently, modal-decomposition-based statefeedback was proposed in [16] for global stabilization of heat equation and in [19] for regional stabilization of Kuramoto-Sivashinsky equation.…”