Exact Controllability Of Wave Equations With Variable Coefficients

Wave equation is an important parabolic Partial Differential Equation . It is usually applied to both vibrations and transmission of waves. It is very helpful for mathematical physics equations to study wave equation. So researching the exact controllability of wave equation is quite useful. In [2],J.L.Lions firstly gives a method for researching the exact controllability of wave equation, a.e. HUM (Hilbert Unique Method ).In this paper,we consider the following system with variable coefficients depending on time by HUM .It is proved that the system is exactly controllable if a(t) ≥ m > 0, a'(t) ∈ L~∞(IR). For the generally case of system with variable coefficients depending on both time and space,we can see [5]. [3] give the exact controllability of systemWe will take the method of [4] and [7] together for researching the controllability in this paper.Firstly,we will prove the existence of the solution of the system.Secondly,we are going to show that the observability inequalityis true if and only if the system is exactly controllable.Finally,the observability inequality will be proved by Multiplier Method and the energy E(t) will be controlled by Gronwall's inequality.