Local null-controllability of a two-parabolic nonlinear system with coupled boundary conditions by a Neumann control
Résumé
This article is concerned with the local boundary null-controllability of a 1-D system of two-parabolic nonlinear equations (often referred as reaction-diffusion system) with coupled boundary conditions by means of a scalar control. The control force is exerted on one of the two state components through a Neumann condition at the left end of the boundary while the other component simply satisfies the homogeneous Neumann condition at that point. On the other hand, at the right end of the boundary, the states are coupled through the so-called $\delta^\prime$-type condition. Upon linearization around the stationary point $(0,0)$, we apply the well-known {\em moments method} to prove the global null-controllability of the associated linearized system with explicit control cost $Me^{M/T}$ as $T\to 0^+$. Then, we show the local null-controllability of the main system by employing the source term method developed in \cite{Tucsnak-nonlinear} followed by the Banach fixed point theorem.
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