BILINEAR CONTROL OF A DEGENERATE HYPERBOLIC EQUATION
Résumé
We consider the linear degenerate wave equation, on the interval (0, 1)
wtt -(x α wx)x = p(t)µ(x)w, with bilinear control p and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory, the "ground state".
Under some classical and generic assumption on µ, we prove that there exists a threshold value for time, Tα = 4 2-α , such that the reachable set is
and a neighborhood of the ground state if α ∈ (1, 2) if T = Tα, the case α = 1 remaining open. This extends to the degenerate case the work [K. Beauchard, Local controllability and non-controllability for a 1D wave equation with bilinear control,
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