Flatness for a strongly degenerate 1-D parabolic equation
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We consider the degenerate equation $$\begin{aligned} \partial _t f(t,x) - \partial _x \left( x^{\alpha } \partial _x f \right) (t,x) =0, \end{aligned}$$on the unit interval \(x\in (0,1)\), in the strongly degenerate case \(\alpha \in [1,2)\) with adapted boundary conditions at \(x=0\) and boundary control at \(x=1\). We use the flatness approach to construct explicit controls in some Gevrey classes steering the solution from any initial datum \(f_0 \in L^2(0,1)\) to zero in any time \(T>0\).KeywordsPartial differential equationsDegenerate parabolic equationBoundary controlNull-controllabilityMotion planningFlatnessReferences1.Abramowitz M, Stegun I (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards. Appl Math Ser 552.Alabau-Boussourira F, Cannarsa P, Fragnelli G (2006) Carleman estimates for degenerate parabolic operators with applications to null controllability. 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