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Title: Boundary Stabilization of the Modified Generalized Korteweg-de Vries-Burgers Equation
Authors: Ala' Waddah Alalabi 
Supervisor: Prof. Nejib Smaoui
Keywords: Burgers Equation , Modified Generalized Korteweg-de Vries
Issue Date: 2018
Publisher:  Kuwait university - college of graduate studies
Abstract: In this thesis, we study the linear and the nonlinear control problem of the Modified Generalized Korteweg-de Vries-Burgers (MGKdVB) equation: u_t+γ_1 u^α u_x-vu_xx+μu_xxx+γ_2 u_xxxx=0, x∈(0,1), and t>0, where v,μ,γ_1 and γ_2 are positive real numbers and α is a positive integer. First, we consider the linear boundary stabilization problem of the MGKdVB equation subject to u(0,t)=0,u(1,t)=0,u_xx (0,t)=0 and u_xx (1,t)=w(t). In the last boundary condition, w(t) plays the role of the input control. In this case, the existence and uniqueness of a global solution are proved, and the exponential stability of the MGKdVB equation in the L^2-sense is established. In addition, we propose a linear adaptive boundary control law for the MGKdVB equation when both of γ_2 and μ are unknowns. Then, we study the nonlinear non-adaptive and adaptive boundary control problems of the MGKdVB equation subject to u(0,t)=0, u_xx (0,t)=0, u_x (1,t)=w_1 (t) and u_xx (1,t)=w_2 (t), where w_1 (t) and w_2 (t) present the nonlinear boundary control. We propose two nonlinear non-adaptive boundary control laws (i.e., when v,μ,γ_1,γ_2 are known) for the equation above. Furthermore, different nonlinear adaptive boundary control schemes are proposed for the MGKdVB equation when one of the parameters v or γ_1 is unknown, when both v and γ_1 are unknowns and finally when all the parameters are unknowns. Using Lyapunov theorem, we investigate the global exponential stability of the solution in L^2 (0,1) for each of the proposed controllers. Last but not least, we support our analytical work with numerical simulations that are in a good agreement with the theoretical results.
Appears in Programs:0410 Mathematics

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