Exact Controllability And Exponential Stability Of A Class Of Wave Equation

In this paper,we deal with exact controllability and exponential stability of the follow wave equationFirstly,by introducing fundamental space and distribution,(1) is equivalent to the following systemSecondly,by introducing the appropriate state space X,input operator B and output operator C,(2) can be rewrite as the state linear system∑(A,B,C,D) on the Hilbert spaceLet A be the infinitesimal generator of a c₀-semigroup on the state space X.Input operator B and output operator C are unbounded linear operators.And,(3) is proved that it is a well-posed linear system.Then,the conservative of(3) is proved.At last exact controllability and exponential stability of(1) are proved by the paper [24].