The Construction Of Solutions For Nonlinear Wave Equation And The Related Control Problem

In this paper.we will mainly study the optimal control problems of the nonlinear wave equations. In first chapters, we will make a simple summary of the optimal control problems and the consistency of the traveling wave solutions of nonlinear equations. The main results will be given in chapters 2 and 3.Partial differential equations are always found in the physics, mechanics, the engi-neering technology and other subjects, one of the important class in partial differential equations is the wave equation, which mainly describes the fluctuation phenomena in nature, including transversal wave and longitudinal wave. The light wave and the sound wave, as well as the galaxy shock wave and the density wave in the universe we are daily familiar with can be all described by the wave equation.In history, many scientists, like J. I. d'Alembert,Euler,D. Bernoulli and Lagrange and so on. had done some significant contributions for wave equation theory when they studied the string vibration problems about musical instrument and other objects.The control theory is a kind of science about the control and communication in studying he machine, it studies the control rule and the information transmission rule between the system's each constituent parts. Based on the analysis of the common characteristics of the automatic control system and the communications system, math-ematician Wiener et al., made the analogies between these systems'control mechanism and some control mechanism in the biology organism, and established this discipline in the 1940s. Since the 1950s when the birth of the Optimal control theory, The research about it has already obtained breaking progress. As early as in the 1950s, people mainly considered the optimal control problems of symposium parameter systems (i.e. systems controlled by ordinary differential equations).In the present paper, we will mainly investigate the existence and other properties of the solution of the wave equation, and then solve the optimal control problemFirst, the consistency of the traveling wave solutions of nonlinear equations are studied by using the Tanh-function method and(G'/G) function expansion method in chapter 2. With deep study on the solutions of nonlinear equations, the research of the traveling wave solutions has become more and more important. It has been applied into many fields such as physics, chemistry, biology and so on, which makes it play an important role in applied science. The mostly used methods to obtain the traveling wave solutions of nonlinear equations are Tanh-function method and(G'/G) function expansion method. In the second and third sections of chapter 3, the principles and procedures of the Tanh-function method and(G'/G) function expansion method are described and analyzed in detail respectively. Such analysis can make us find their common parts. Considering the general nonlinear wave equation, we use Tanh-function method and(G'/G) function expansion method to solve their traveling wave solutions respectively.In Chapter 3 we study the optimal control of the nonlinear one dimensional pe-riodic wave equation with.x-dependent coefficients.First,we establish the properties of wave equation with x-dependent coefficients.Then we obtain the existence and regular-ity of periodic solution.Last, the existence of the optimal control is proved by means of Arzela-Ascoli lemma and Sobolev compact imbedding theorem.In the chapter, we conside wave equation: and establish the properties of wave equation with x-dependent coefficients.In the sec-ond sections of chapter4, we give the definition of weak solution:Definition 0.1 The function y∈L2(Q)is said to be a weak solution to (3.2.1),ifφ∈X,y satisfy whereSet D(A)={y∈L2(Q):there is f∈L2(Q)such that (0.0.7) holds}. Define A:D(A)→L2(Q) by ifIn terms of A,the weak solution y to (3.2.1)is the solution to operator equation Ay=u-1f, Note that for each y∈D(A) there is a precisely one f∈L2(Q) such that Ay=u-1f (due to the density of X in L2(Q)), so A is a well-defined linear operator.We have suppose as:Then we give the properties of wave operator with x-dependent coefficients:Theorem 0.2 Assume that T=p/q and u∈H2(0,1) satisfies (H1), (H2). Then A is a closed operator with a closed range R(A)=N(A)⊥, A is self-adjoint and A-1∈ L(R(A),R(A)).In addition,we have the following estimates where d=inf{|λn2-μm2|,λn≠|μm|},α=inf{μm2-λn2,|μm|>λn},and C is a constant.In third section we use the propert of operater A,we obtain the existence and regularity of solution to state equation(3.2.1).Last,the existence of the optimal control is proved by means of Arzela-Ascoli lemma and Sobolev compact imbedding theorem. And the result can be given as follows:Theorem 0.3 Set T=p/q and u∈H2(0,1) satisfies (H1),(H2). In addition, (H3) holds. Then, The optimal control problem (3.1.1-3.1.2) has at least one optimal solution g*∈∑.