Exact Controllability Of The Equation With Rotation Control

The paper studys exact controllability of the linear beam equation with rotationcontrol. Uses the method is literature [1] studies the wave equation exact controllabilitymethod, a.e. HUM(Hilbert Unique Method).The main conclusion in this paper is that the following systemis proved exactly controllable in state space L²(Ω)×H^-2(Ω) if T＞2R/μ₁^1/2.Firstly, we will prove the existence of the solution for the following homogeneousbeam equation.Furthermore, the solution satisfies the formulaΔu(t)∈L²(0, T; L²(Γ)).Secondly, we are going to show the existence of the solution of the beam vibrationsystem.Finally, it will be proved that the system is exactly controllable if and only if theHilbert space H₀²(Ω)×L²(Ω) and the duality space L²(Ω)×H^-2(Ω) are isomorphicspaces.