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A numerical approach to the exact boundary controllability of the wave equation. I: Dirichlet controls: Description of the numerical methods. (English) Zbl 0699.65055
Numerical aspects of the control problem which consists in driving to rest the solution of the wave equation are considered.
This problem is reformulated as a linear variational problem and solved by a conjugate gradient iteration. A finite element approximation of the continuous case is introduced and specified for the unit square as spatial domain. For testing, an exact solution is constructed.
Preliminary computational results exhibit good behaviour only for coarse spatial discretization. As the source of growing ill-posedness the approximation of the normal derivative is identified. Tikhonov regularization (with heuristically and computationally justified regularization parameter) helps to battle this difficulty.
The paper concludes with a discussion of the minimal control time and with a lot of graphically presented numerical results which have been obtained by spending at least one but probably several hours on a supercomputer.
Reviewer: G.Stoyan

MSC:
65K10 Numerical optimization and variational techniques
93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
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