On Controllability And Asymptotic Stability Of Nonlinear Waves

Control engineering is a large subject. Optimal control engineering of nonlinear waves often assumes that obtaining a model of the nonlinear wave systems is available and one wants to optimize its behavior. Nonlinear wave models established from the control engineering vary according to the applications: for example, some ordinary differential equations are used when modelling a continuous signal, various partial differential equations are used to model sampled signals and nonlinear waves as used in digital systems or water motion systems. In reasoning about such systems we are interested not only in the solutions, but in their stabilities and long time asymptotic behavior under various assumptions on input initial or input initial-boundary data.Asymptotic stabilization in the branch of control engineering has been a subject of active research since 1960's. One of the major motivating factors has been the realization that existing theories on control systems analysis and stability are inadequate for solving modern day problems in nonlinear waves and a variety of complex nonlinear systems. In order to solve these complicated problems in control engineering systems, controlling the long time asymptotic behavior of nonlinear waves needs a lot of advanced mathematical tools and techniques so as to achieve a desired goal. These lead the development of a rich theory of nonlinear control systems analysis. Controllability of linear and nonlinear wave systems represented by nonlinear wave equations has been studied by a number of authors. This concept has been extended to discuss the existence and asymptotic stability of global solutions in time in various Banach spaces, which include Sobolev and classical spaces. Some approximate controllability of semilinear control systems using fundamental assumptions on the systems components havebeen established. The asymptotic theory in control engineering for hyperbolic partial differential equations in some Sobolev spaces with several controlled initial or initial-boundary assumptions is key to analyze the phenomena existing in nature.It is worth to notice that all the nonlinear waves considered in this paper are related to some nature phenomena that some experts have researched in the field of control engineering(see [14-17]). Our results about the stability and long time asymptotic behavior of nonlinear waves are in some sense different from theirs. We consider the nonlinear Boussinesq waves and other two nonlinear waves in the Sobolev space or classical space.The purpose of the present work is to obtain some results involving the well-posedness and asymptotic stability of global solutions for three generalized Boussibesq equations and two nonlinear wave equations with some controlled assumptions on the initial data or initial-boundary data. Also, it is due to the fact that the solutions of generalized Boussinesq equations and nonlinear wave equations discussed in the work represent the solitary waves, we have got that solitary waves obtained in the paper are stable.This work constitutes mainly six parts.1: The background and significance of this project are given.2: The asymptotic stability of controlled initial values for a generalized Boussinesq equation is given.The asymptotic behavior of this Boussinesq wave is analyzed.3: We have discussed the globally stability of a controlled initial-boundary problem for the damped boussinesq equation. The methods to calculate the asymptotic solution is established.4: The stability of solitary waves for a generalized Boussinesq equation is obtained.5: We have got that the stability of global solution for a damped Euler-Bernoulli equation in a minus index Sobolev space.6: The asymptotic theorem of global solutions for semilinear wave equations with nonspheral symmetric forms in three space dimensions is established. As an application of the asymptotic theory, a special wave equation is analyzed in detail.