Exact Controllability Of Beam Equations

In this paper,we study the exact controllability of the beam equationswhether the beam equations (*) is exactly controllable or not,we impose the controlv(t) on the boundary y(L,t).In the studying course of the exact controllability of the beam equations (*),firstly,we consider the following homogeneous beam equations:we deduce some results from the beam equations (**);then we obtain a map on thetwo beam equations in the zero initial data;finially,we show the surjectivity of themap. Therefore we obtain the following result:If T＞L/(μ1)^1/2, then for any given (y⁰, y¹)∈H^-1[0, L]×H^-3[0, L], there existsv(L,t)∈L²(0, T), such that the solution of (*) satisfies:That is the beam equations (*)is exactly controllable.