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This work is concerned with the necessary conditions of optimality for aminimal time control problem $(P)$ for the linearized Navier-Stokes periodicflow in a 2D-channel, subject to a boundary input which acts on the transversalcomponent of the velocity. The objective in this problem is the reaching of thelaminar regime in a minimum time, as well as its preservation after this time.The determination of the necessary conditions of optimality relies on theanalysis of intermediate minimal time control problems $(P_{k})$ for theFourier modes $"k"$ associated to the Navier-Stokes equations and on the proofof the maximum principle for them. Also it is found that one can construct, onthe basis of the optimal controllers of problems $(P_{k}),$ a small time calledhere \textit{quasi minimal} and a boundary controller which realizes therequired objective in $% (P).$

Exact controllability in minimal time of the Navier-Stokes periodic flow in a 2D-channel