一类非线性发展方程的边界自适应稳定
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摘要
本论文主要研究了一类非线性偏微分方程：粘性Burgers方程、广义粘性Burgers方程和KdVB方程的边界自适应控制的问题以及时滞的KdVB方程的稳定性问题。在适当的边界条件下,通过选取合适的的自适应边界控制律,得到了粘性Burgers方程、广义粘性Burgers方程和KdVB方程的稳定性和收敛性结果。并证明了所讨论的系统在所给边界条件下解的存在性和唯一性。
在第一章中,我们首先介绍了粘性Burgers方程、KdV方程以及KdVB方程的物理背景以及边界控制和自适应控制的研究进展,其次介绍了研究稳定性的主要方法李雅普诺夫方法以及在自适应稳定的判断中经常用到的Barbalat引理,最后,我们给出在本论文的研究中经常用到的一些相关的定义、定理和不等式。
在第二章中,我们研究了粘性Burgers方程的边界自适应控制的问题。在所给边界条件下,证明粘性Burgers方程的解是全局L2-稳定和全局H’-稳定的。利用Barbalat引理,证明了系统的解的L2收敛性和H1收敛性。最后,利用格林函数证明了粘性Burgers方程在所给边界条件下解的适定性。
在第三章中,我们研究了广义粘性Burgers方程的边界自适应控制的问题。在所给边界条件下,证明广义粘性Burgers方程的解是全局L2-稳定和全局H1-稳定的。耗散项的增加也增加了估计的难度,所以利用Barbalat引理,我们得到了系统的解的L2收敛性。最后,利用格林函数的方法证明了广义粘性Burgers方程在所给边界条件下解的适定性。
在第四章中,我们研究了KdVB方程对于未知的色散系数的边界自适应控制的问题。在所给边界条件下,证明KdVB方程的解是全局L2-稳定的。利用Barbalat引理,得到了系统的解的L2收敛性。最后,利用Galerkin方法和Banach压缩不动点定理证明了KdVB方程在所给边界条件下解的适定性。
在第五章中,我们研究了广义KdVB方程的非自适应控制和自适应控制的问题。在非自适应控制的情形下,证明广义KdVB方程的解是全局指数稳定的。在自适应控制的情形下,得到了系统的解的L2收敛性。
在第六章中,我们研究了时滞的KdVB方程解的渐近行为。利用算子半群的理论证明了时滞的KdVB方程解的存在性和唯一性。如果时滞项τ足够小的话,我们证明了时滞的KdVB方程的解是指数稳定的。最后,第七章是本论文的结论。
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.
在第一章中,我们首先介绍了粘性Burgers方程、KdV方程以及KdVB方程的物理背景以及边界控制和自适应控制的研究进展,其次介绍了研究稳定性的主要方法李雅普诺夫方法以及在自适应稳定的判断中经常用到的Barbalat引理,最后,我们给出在本论文的研究中经常用到的一些相关的定义、定理和不等式。
在第二章中,我们研究了粘性Burgers方程的边界自适应控制的问题。在所给边界条件下,证明粘性Burgers方程的解是全局L2-稳定和全局H’-稳定的。利用Barbalat引理,证明了系统的解的L2收敛性和H1收敛性。最后,利用格林函数证明了粘性Burgers方程在所给边界条件下解的适定性。
在第三章中,我们研究了广义粘性Burgers方程的边界自适应控制的问题。在所给边界条件下,证明广义粘性Burgers方程的解是全局L2-稳定和全局H1-稳定的。耗散项的增加也增加了估计的难度,所以利用Barbalat引理,我们得到了系统的解的L2收敛性。最后,利用格林函数的方法证明了广义粘性Burgers方程在所给边界条件下解的适定性。
在第四章中,我们研究了KdVB方程对于未知的色散系数的边界自适应控制的问题。在所给边界条件下,证明KdVB方程的解是全局L2-稳定的。利用Barbalat引理,得到了系统的解的L2收敛性。最后,利用Galerkin方法和Banach压缩不动点定理证明了KdVB方程在所给边界条件下解的适定性。
在第五章中,我们研究了广义KdVB方程的非自适应控制和自适应控制的问题。在非自适应控制的情形下,证明广义KdVB方程的解是全局指数稳定的。在自适应控制的情形下,得到了系统的解的L2收敛性。
在第六章中,我们研究了时滞的KdVB方程解的渐近行为。利用算子半群的理论证明了时滞的KdVB方程解的存在性和唯一性。如果时滞项τ足够小的话,我们证明了时滞的KdVB方程的解是指数稳定的。最后,第七章是本论文的结论。
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.
引文
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