Can Be Controlled One Dimension Wave Equation With Mixed Boundary Problem

In this paper, we concerned with the following wave equationFirst, we proved that there exists a solution for the problem (0.1) under some conditions.Then we studied the solution energy to discuss the relation controlling problem.(1)For giving time,there exists an interval such that we can control the solution energy on the interval.(2)For giving an interval,there exists a time such that we can control the solution energy at any time.It is difficult to study these problems.First,we used the method of separation of variables to compute the problem with Fourier series solution,then wo found theboundary conditions for functions (?)₁ (x), (?)₂ (x),ψ₁ (x) andψ₂ (x) .At least, we computedthe energy of the solution and illustrated the applications in Physics.We used the superposition principle (the properties of linear homogeneous equations) to divide the problem (0.1) to the following two problems (0.2),and(0.3)AndUsed the mothod of separation of variables to get the solutions of (0.2) and (0.3), then get the solution of (0.1), at least to discuss the energy of the solution and get our results.