The Exact Controllability Of Wave Equation With Dirichlet Boundary

consider the exact controllability of wave equation with Dirichlet boundarywhere Ω is the region with bound in Rⁿ. the superscript"'" denote differentaion with respect to the variables t. "A" is laplace operator, q e L^∞(Ω ) and q≥ 0. v is control function,and v ∈ L_loc²(R;_loc²(Γ₁)). The exact controllability of the system was proved by V.Komornik by using HUM(hilbert uniqueness method) in paper[3] in 1994, when where λ₁ is the biggest constant such thatIn this paper,The exact controllability of the system is obstained by HUM(hilbert uniqueness method) and the knowledge of Riemannian manifold, when positive eigenvalue of all the eigenvalue corresponding the operater "—A" In paper[3] the equality (18) is obstained by (16)+(n-l)(17)i.e.But in this Paper, the equality (2.10) which is similar to (18) in paper[3] is gained by transition in (2.8) by useing the equality (2.9)namelynot only the considering in terms of the value of "n" is precluded,but also the equlity (3.10) is better estimative than the equlity (18) in paper[3]. consider the exact controllability of wave equation with dampThe exact controllability of the system was proved by sunbo in 2002 when the control time "T" long enough and the damp coefficient "k" is small enough. But the range "T"and "k" weren't given clearly,in this paper the proof is deduced again and the range "T"and "k" is obstained.i.e.when k < j^pr-, T > 2fc(""^_nfc ,the system (2) is exactcontrollable. Because exact controllability of the system on basis of the existence of solution so the uniqueness and existence of solution is considered in chapter 2 . in some paper the proof of controllability is completed when the control function make the solution of system is null in time " T". in fact, only the The exact null controllability of the system was proved,it is difficut to understand to new learner,so in chapter 2, not only the defination of the exact null controllability and the exact controllability is given but also the relation of equality between the exact null controllability and the exact null controllability is proved. Then the observability of system is considering,i.e.there are positive fixed constant "ci","c2",such thatClï¿¡(0) < f f (⁾2dTdt < c2E(0).Jo Jr0 °vIn chapter 3 because the range of control time "T"and the damp coefficient "k" in paper[3] wasn't given clearly,so the proof is deduced again by HUM(hillbert unique methord) and the knowledge of Riemannian manifold and the range "T"and "k" is obstained,so the result of paper[12] is improved .