Local stabilization of semilinear parabolic system by boundary control

Abstract

This paper addresses local stabilization of a semilinear parabolic system by boundary control. Under a certain assumption on the nonlinearity of the parabolic equation, a linear boundary feedback control law is applied to control the semilinear system. Based on the operator theories and relations of different norms, locally exponential stabilization of the closed loop system is established. Simulations support the established stability result.

Local stabilization of the Burgers–Fisher equation via boundary control

“…With respect to general nonlinear parabolic PDEs, control laws for global stabilization remain open problems [11][12][13][14][24][25][26]. Currently, there exist mathematical dif-ficulties to construct control laws for global stabilization of general nonlinear PDEs [11][12][13][14][24][25][26]. To solve this problem, scholars have to settle for second best and establish control laws for the local stabilization of these systems.…”

“…To solve this problem, scholars have to settle for second best and establish control laws for the local stabilization of these systems. For example, [11] discussed the stabilization problem for a one-dimensional nonlinear Fisher's PDE defined on a bounded interval, [12] discussed the local exponential stabilization of a class of semilinear parabolic systems, [13] addressed the stabilization problem for the nonlinear Korteweg-de Vries equation posed on a bounded interval, and [14] discussed the local exponential stabilization of coupled nonlinear parabolic equations. However, for certain special nonlinear PDEs, for example, the parabolic PDE with Volterra nonlinearity, it was still feasible to construct a boundary control law for global stabilization [15,16].…”

“…Inspired by [11][12][13][14], we study the local exponential stabilization problem for the nonlinear Burgers-Fisher (BF) equation in this paper. Although this equation has been used widely in fluid dynamics, aerodynamics, nonlinear optics, chemistry, and biology, its stability has not been resolved until now.…”

“…In fact, a few nonlinear PDEs allow the system to be globally stabilized by constructing a boundary feedback control law [15,16]. Currently, there exist mathematical difficulties to construct control laws for global stabilization of general nonlinear PDEs [11][12][13][14][24][25][26]. However, it is practical to establish control laws for local stabilization.…”

Results on Boundary Control for Parabolic Systems Using Backstepping Method

“…e boundary control for parabolic PDEs [4][5][6], for hyperbolic PDEs [7,8] and especially for nonlinear PDEs [9][10][11] is widely studied using the backstepping method. is method is introduced in [12,13] by constructing an integral operator which maps the solution of heat equation onto solution of linear parabolic equation with analytical coefficients.…”