Reduced-Order Based Distributed Feedback Control Of The Benjamin-Bona-Mahony-Burgers Equation

With the development of science and technology,nonlinear problems of equations play more and more important roles in the fields of natural science and social science.There are many nonlinear problems in physics,chemistry,engineering,biology,eco-nomics and other fields.To solve these nonlinear problems,it is necessary to establish nonlinear partial differential equations to solve them.In this paper,we study a class of very important nonlinear partial differential equations,Benjamin Bona Mahony Burg-ers(BBMB)equation.In physics,the dispersion.effect of BBMB equation is the same as that of BBM Equation,but the dissipation effect is the same as that of Burgers equation.In physical sense,the equation with dissipation term can well predict and explain the phenomena of water wave and aperture propagation.Meanwhile,BBMB equation is an alternative model of Korteweg de Vries burgers(KdVB)equation.In recent years,many researchers have studied the behavior of the solution of BBMB equation in mathematical and physical sense,and discussed the numerical solution of BBMB equation by using finite difference,finite element or domain decomposition method.This paper mainly discusses the reduced-order model of BBMB equation and its application in distributed feedback control.The study of BBMB equation space is based on the finite element method(FEM)using B-spline function approximation.Firstly,we find the form of the BBMB equation by using the formula of the partition,and then derive the form of the semi discrete according to the semi discretization of the space domain.Secondly,we construct the quadratic spline function to approximate.Then we write the equation system as a nonlinear ordinary differential equation with N equations and N unknowns.Finally,we get the numerical solutions from the starting point of the equilibrium solution.We use these solutions to generate the snapshots set.In order to determine the basis function of the approximation subspace,we need to explain the centroidal Voronoi tessellation(CVT)method which uses the snapshots as the basis of order reduction.Firstly,Voronoi region is introduced,and the defi-nition of centroid is given.When the generator of Voronoi mosaic region coincides with the centroid,the mosaic is called centroidal Voronoi tessellation(CVT),and the optimization characteristics of CVT are discussed.Then we introduce another method to construct CVT basis,that is to use coefficient vector of finite element function.For comparison,this paper briefly introduces the proper orthogonal decomposition(POD)method.POD method also starts from a group of snapshots vectors,and then con-structs snapshots matrix.It can calculate singular value decomposition of matrix to find POD base,and can also calculate eigenvalue of correlation matrix to determine POD base.Then some properties of POD bases are introduced.We can also construct POD bases based on functions rather than vectors.With the reduced basis,we can build the reduced-order modeling of BBMB equation.Of course,there will be some errors in any solution method.We analyzed the source of the errors,and compared the two reduced-order modelings through the specific calculation and analysis of numerical experiments.We concluded that POD and CVT have similar accuracy.The last part of the paper introduces the feedback control system of BBMB e-quation.The application in distributed feedback control system is mainly realized by centroidal Voronoi tessellation(CVT).In the end,we have carried on the calculation experiment to the specific example of BBMB equation,compared the uncontrolled so-lution with the full order controlled solution and the reduced order controlled solution,the obtained solutions are very stable,so the full-order controlled solution and the reduced-order controlled solution have almost the same effect.