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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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