Local stabilization of coupled nonlinear parabolic equations by boundary control

Local stabilization of semilinear parabolic system by boundary control

“…The difficulty of this work is the stability analysis of the closed loop system. The nonlinearity of this problem not only depends on x , t , u but also u x , which implies that system () represents more widely class of nonlinear problems in comparison with [20]. Besides, this work may encourage someone to further explore new methods for nonlinear problems.…”

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.…”

Boundary Stabilization of Heat Equation with Multi-Point Heat Source

“…Actually, the proposed finite-dimension feedback linearization method was infinite dimensional extension of the backstepping approach. It is well known that backstepping transformation is mainstream method to deal with boundary stabilization problems of parabolic PDEs, such as, linear parabolic PDEs [15][16][17][18][19], nonlinear parabolic PDEs [20][21][22], quasi-linear parabolic PDEs [23], coupled parabolic PDEs and ODE [24][25][26][27].…”