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We study a class of Lindblad equation on finite-dimensional fermionicsystems. The model is obtained as the continuous-time limit of a repeatedinteraction process between fermionic systems with quadratic Hamiltonians, asetup already used by Platini and Karevski for the one-dimensional XY model. Weprove a necessary and sufficient condition for the convergence to a uniquestationary state, which is similar to the Kalman criterion in control theory.Several examples are treated, including a spin chain with interactions at bothends.

Repeated interaction processes in the continuous-time limit, applied to quadratic fermionic systems