Abstract
This work is concerned with the necessary conditions of optimality for a minimal time control problem $(P)$ related to the linearized Navier--Stokes periodic flow in a 2D-channel, subject to a boundary input which acts on the transversal component of the velocity. The objective in this problem is reaching the laminar regime in minimum time and preserving it after this time. The determination of the necessary conditions of optimality relies on the analysis of intermediate minimal time control problems $(P_{k})$ for the Fourier modes $``k''$ associated to the Navier--Stokes equations and on the proof of the maximum principle for them. Also it is found that one can construct, on the basis of the optimal controllers of problems $(P_{k}),$ a small time, called here quasi-minimal, and a boundary controller which realizes the required objective in $(P).$
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