Singular Optimal Control for a Transport-Diffusion Equation |
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Authors: | S Guerrero G Lebeau |
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Institution: | 1. Laboratoire Jacques-Louis Lions , Université Pierre et Marie Curie , Paris, France guerrero@ann.jussieu.fr;3. Laboratoire J.-A. Dieudonné , Université de Nice – Sophia Antipolis , Nice, France |
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Abstract: | In this paper we consider a transport-diffusion equation with coefficient of diffusion ε > 0 small and coefficient of transport M(x, t). We study the asymptotic behavior of the cost of the null controllability of such a system when ε![/></span> 0<sup>+</sup>.</p> If at least one trajectory associated to <i>M</i>(<i>x</i>, <i>t</i>) does not enter the control zone, we prove that this cost explodes exponentially as ε<span class=](/na101/home/literatum/publisher/tandf/journals/content/lpde20/2007/lpde20.v032.i12/03605300701743756/production/images/medium/lpde_a_274330_o_ilf0002.gif) ![/></span> 0<sup>+</sup>. On the other hand, as long as trajectories reach the control region and the controllability time is sufficiently large, we prove that the cost is bounded as ε<span class=](/na101/home/literatum/publisher/tandf/journals/content/lpde20/2007/lpde20.v032.i12/03605300701743756/production/images/medium/lpde_a_274330_o_ilf0002.gif) ![/></span> 0<sup>+</sup>, and moreover decays exponentially as ε<span class=](/na101/home/literatum/publisher/tandf/journals/content/lpde20/2007/lpde20.v032.i12/03605300701743756/production/images/medium/lpde_a_274330_o_ilf0002.gif) ![/></span> 0<sup>+</sup> as soon as all trajectories cross the boundary.</td>
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Keywords: | Carleman estimates Controllability Heat equation Singular limits |
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