Controllability And Stability Analysis Of Some Hyperbolic Equations

Partial differential equation(PDE)is an important mathematics branch.In biology,physics,chemistry,engineering and other modern natural sciences,scientists often use PDEs to model some practical problems.Controllability and stability have been the problems that have been widely concerned.The controllability and stability of some hyperbolic equations are mainly studied in this thesis.For the controllability problem of hyperbolic equations,we mainly discuss the simultaneous controllability of two nonlinear wave equations.For the stability of hyperbolic equation,we mainly study the energy decay of a plate equation with local linear damping in the exterior domain and the energy decay of a one-dimensional wave equation in R with nonlinear damping.The thesis consists of four chapters.In Chapter 1,we provide a simple research summary of the stability and controllability of hyperbolic equation and introduce the main research problems of this paper.In Chapter 2,we consider the following plate equations with local damping ???where a(x)is a positive function and its support set satisfies certain conditions.By defining the weight energy and using the multiplier method,an estimation formula containing the weight energy is obtained.Finally,the decay property of the original energy of the system is obtained by proving the boundedness of the weight energy.In Chapter 3,we consider the simultaneous controllability of two nonlinear wave equa-tions???where f(x)? C1(R)is a nonlinear function and satisfies the local Lipschitz condition.First,we use HUM(Hilbert uniqueness theorem)to prove the simultaneous controllability of thelinearized system of the original system.Then we use fixed point theorem to prove the simultaneous controllability of the nonlinear system.In Chapter 4,we consider the following wave equation with nonlinear damping???where the initial values of the above system satisfy some conditions.By constructing a proper truncation function,the above system can be transformed into a wave equation on the exterior domain with Dirichlet boundary conditions under certain conditions.Then we construct an auxiliary function.Finally,the boundedness of its weight energy is obtained and the attenuation characteristics of the original energy is achieved.