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

This work is concerned with the necessary conditions of optimality for a minimal time control problem $(P)$ for 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 the reaching of the laminar regime in a minimum time, as well as its preservation 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 \textit{quasi minimal} and a boundary controller which realizes the required objective in $% (P).$