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Exact output tracking for the heat equation.
Abstract
In this talk we will consider the following model: the temperature $z(t,x)$ evolves on a bounded $x$-inteval and we are allowed to choose a pointwise control (e.g. one prescribes the themperature at the left end). The classical problem of control theory is the exact controllability: given a horizon time $T$ and a final state $z^T(x)$, find a control such that the solution satisfies $z(T,x) = z^T(x)$ for all $x$. We will focus on a different problem which is called the output regulation (or sometimes tracking), for which one defines an output $y(t)$ measured from the state $z(t)$ (e.g. a point measurement $y(t) = z(t,x_0)$). Given a reference signal $y_ref(t)$, the objective is to find a control such that the output $y(t)$ of the system matches the reference signal. When the output is a point measurement, such trackable signals have been characterized recently in collaboration with P. Lissy (CERMICS). I will sketch the proof of this result, which relies on Paley-Wiener theory and a new Plancherel type isometry for Gevrey functions. If the time permits, I will also present a consequence of our results on the structure of Gevrey functions.
