Exact Controllability For A Linear Hyperbolic System With Zero Eigenvalues

In this paper,we consider the exact controllability for a linear hyperbolic system with zero eigenvalues.the given initial conditions t = 0:(u,v,w)=(u0,v0,w0} 0 ≤ x ≤ L and final conditions t = T:(u,v,w)=(uT,vT,wT,),0 ≤ x ≤ L L is the length of[0,L].There are two main parts.The first part,we give a anthropogenic coupling relationship with the boundary conditions.In this case,we discussed the mixed initial-boundary value problem of exact controllability.First of all,we get the necessary and sufficient condition(only related to the initial conditions and final conditions)to achieve purpose just by the exact boundary controllability.When the necessary and sufficient condition satisfied,we can achieve controllability respectively by the two sides boundary control and two sides boundary internal control.Then,considering the case that the necessary and sufficient condition does not meet,we impose appropriate internal control on nonzero eigenvalues corresponding to the equations together with properly boundary control we realize the exact controllability.The second part,in order to realize one side exact controllability,we give a bound-ary condition with a coupling relationship,and then we realize the exact controllability of the mixed initial-boundary value problem.when the sufficient and necessary con-ditions we discussed at the first part satisfied,we can realize the one side and two side boundary exact controllability.And then we discuss the case the necessary and sufficient condition does not meet,we realize the exact controllability through one side and two sides boundary-internal control.