| Abstract: |
| In this talk, I will address an exact output tracking problem (or sidewise control problem) for the one-dimensional heat equation on a bounded interval, with Neumann boundary conditions. The control is a Neumann control on the left of the interval, whereas the output is the Dirichlet trace at the right-hand side of the interval. We aim at characterizing all the possible outputs when the control $u$ lives in the space $L^2(0,T)$ for a finite or infinite $T$. In this setting, we obtain an exact characterization of all the trackable outputs. In infinite time, they form some kind of Gevrey class, whereas in finite time, some auxiliary power series involving the derivative of the signals in time T needs also to belong to the reachable space. The proof is based on applying the Laplace transform in time , and notably on some auxiliary lemmas on the Hardy spaces that might be interested by themselves, and some kind of Plancherel formula. If time permits, I will explain the link with some classical problems in real analysis, namely, the interpolation problems in some Gevrey classes. |
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