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Small time global null controllability for a viscous Burgers’ equation despite the presence of a boundary layer. (English) Zbl 1291.93043
Summary: In this work, we are interested in the small time global null controllability for the viscous Burgers’ equation $$y_t-y_{xx}+yy_x=u(t)$$ on the line segment $$[0,1]$$. The right-hand side is a scalar control playing a role similar to that of a pressure. We set $$y(t,1)=0$$ and restrict ourselves to using only two controls (namely the interior one $$u(t)$$ and the boundary one $$y(t,0)$$). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole-Hopf transform and Fourier series to derive precise estimates for the creation and the dissipation of a boundary layer.

##### MSC:
 93B05 Controllability 93C20 Control/observation systems governed by partial differential equations 35Q53 KdV equations (Korteweg-de Vries equations)
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