Regularization Methods For The Boundary Controllabilities With The Heat Equation

Because of the technology and physical, for some systems, it can only put controldevice set on the edge of the area. In the recent decades, due to getting the attention of thecontrol theory, the boundary control problem have a great development. Now, more andmore people focus on the boundary control of the heat equation, the KDV equation and soon, and acquired some achievements.In this paper, we mainly study the problem that the exact boundary controllability ofthe heat equation. As is known to all, the problem is not exactly controllable, so we reducethe problem to the ill-posed inverse problem. we try to choose T ikhonov regularizationmethod to regularize it, and use GCV method to choose the regularize parameterOn the one hand, first, we reduce the exact boundary controllability problem to theapproximate boundary controllability problem of the heat equation. Then, through Hilbertuniqueness method, using convex duality, we reduce the solution of the boundary controlproblems to the solution of identification problems for the terminal data. In general, theproblem is ill-posed. Then, we regularize it by Tikhonov method, and give the numericalresults.On the other hand, through the operator that is given by the heat equation, we definethe characteristic vector as an orthonormal basis. The initial state and the terminal statecan express by the orthononormal basis. Through the solution of the adjoint equation, wereduce the exact boundary controllability problem to the Fredholm equation. Then, weregularize it by Tikhonov method, and give the numerical results.