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In this paper, we introduce the numerical backstepping method by applying it to a problem of finite-time stabilization for a system of   \begin{document}$ 2 \times 2 $\end{document}   balance laws discretized thanks to the upwind scheme. On the one hand, we illustrate on an example that the scheme used to compute the feedback control cannot be chosen arbitrarily. On the other hand, an algorithm is given to construct this control properly and an approached finite-time stabilization result is proven.

The discretized backstepping method: An application to a general system of $ 2\times 2 $ linear balance laws