Approximate control of parabolic equations with on-off shape controls by Fenchel duality
arxiv(2023)
摘要
We consider the internal control of linear parabolic equations through on-off
shape controls, i.e., controls of the form M(t)χ_ω(t) with M(t)
≥ 0 and ω(t) with a prescribed maximal measure. We establish
small-time approximate controllability towards all possible final states
allowed by the comparison principle with nonnegative controls. We manage to
build controls with constant amplitude M(t) ≡ M. In contrast, if the
moving control set ω(t) is confined to evolve in some region of the
whole domain, we prove that approximate controllability fails to hold for small
times. The method of proof is constructive. Using Fenchel-Rockafellar duality
and the bathtub principle, the on-off shape control is obtained as the
bang-bang solution of an optimal control problem, which we design by relaxing
the constraints. Our optimal control approach is outlined in a rather general
form for linear constrained control problems, paving the way for
generalisations and applications to other PDEs and constraints.
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