Boundary Adaptive Stabilization Of A Class Of Nonlinear Evolution Equation

In this doctoral dissertation, we study the adaptive boundary control of the viscous Burgers equation, generalized viscous Burgers equation, Korteweg-de Vries-Burgers and generalized Korteweg-de Vries-Burgers. Under proper boundary conditions,Using appropriate adaptive boundary control law, we get the stability and the regulation of the nonlinear Burgers equation, generalized viscous Burgers equation, Korteweg-de Vries-Burgers and generalized Korteweg-de Vries-Burgers. We also proved the existence and uniqueness of the above system.The dissertation is divided into six chapters. In the first chapter, we first introduce the puysical background of the equations and the latest research advances of boundary control and adaptive control. Then we introduce the main method-Lyapunov method to analyse the stability of the solutions of the equations and the Barbalat Lemma used to analyse the adaptive stabilization. We then give some related definitions,Theorems, inequalities and notations of the dissertation.In the second chapter, we study the adaptive boundary control of the viscous Burgers equation, Under the given boundary conditions, we proved the L2-stability and the H1-stability of the solution of the viscous Burgers equation. By Barbalat Lemma, we proved the L2-regulation and the H1-regulation of the solution of the viscous Burgers equation. We also proved the existence and uniqueness of the viscous Burgers equation by Green function.In the third chapter, we study the adaptive boundary control of the generalized viscous Burgers equation, Under the given boundary conditions, we proved the L2-stability and the H1-stability of the solution of the generalized viscous Burgers equation. By Barbalat Lemma, we proved the L2-regulation of the solution of the generalized viscous Burgers equation. We also proved the existence and uniqueness of generalized viscous Burgers equation by Green function.In the fourth chapter, we study the adaptive boundary control of the Korteweg-de Vries-Burgers equation when the viscosity coefficient δ is unknown. Under the given boundary conditions, we proved the L2-stability of the solution of the KdVB equation. By Barbalat Lemma, we get the L2-regulation of the solution of the system. Using the Galerkin method and the Bananch contractive fixed point theorem, We proved the existence and uniqueness of the Korteweg-de Vries-Burgers equation.In the fifth chapter, we study the non-adaptive boundary control and the adaptive boundary control of the generalized Korteweg-de Vries-Burgers equation, Under the non-adaptive boundary control, we proved the L2-stability exponentially of the generalized Korteweg-de Vries-Burgers equation, Under the adaptive boundary control, we get the L2-regulation of the solution of the KdVB system.In the sixth chapter, we study the asymptotic behavior of the time-delayed Korteweg-de Vries-Burgers equation. By the operator semigroup theory, we proved the existence and uniqueness of the time-delayed Korteweg-de Vries-Burgers equation. If the time-delay parameter r is small enough, we proved the exponentially stability of the time-delayed Korteweg-de Vries-Burgers equation. The last chapter is the conclusion of the dissertation.