13-08-2012, 03:31 PM
Numerical Approximation of the Exact Control for the String Equation
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Abstract
In this paper we implement the results obtained by Vasilyev et al [11] on the numerical
approximation of the exact control for the string equation. The computational part and the
respective graphs are made for a particular case. For that we have applied the Residues
Theorem of holomorphic functions, which, as far as we know, is the rst time that this
theorem is applied in the computational study of exact control problems.
Computational Results
In this section we determine the graphs of the approximate control uN(t) and of the approximate
ultra weak solution yN(x; t) at the instant t = T. Here y(x; t) denotes the solution of
Problem (5) with T = 3L, exact control u(t) and initial data y0 1 , y1 = L=2 .
By the characteristic of the matrix M dened by (13), the Crout form or (LTDL) (see
Golub et al [3]) is an appropriate method for solving the system, that is, for obtaining the
coecients.
Appendix
In this part by using the Residue Theorem of holomorphic functions we obtain some
results on numerical series. For other similar results, see Weinberger [12].
In what follows we x some notations and write some results. Let O be an open set of
CI and f : O ! CI a holomorphic function. Let z0 be a pole of order k of f(z). The residue
of f(z) at z0, which is denoted by <z0