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Localising optimality conditions for the linear optimal control of semilinear equations \emph{via} concentration results for oscillating solutions of linear parabolic equations

Mazari, Idriss; Nadin, Grégoire (2022), Localising optimality conditions for the linear optimal control of semilinear equations \emph{via} concentration results for oscillating solutions of linear parabolic equations. https://basepub.dauphine.psl.eu/handle/123456789/24089

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Type
Document de travail / Working paper
Date
2022
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
23
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Author(s)
Mazari, Idriss
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Nadin, Grégoire
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Abstract (EN)
We propose a fine analysis of second order optimality conditions for the optimal control of semi-linear parabolic equations with respect to the initial condition. More precisely, we investigate the following problem: maximise with respect to y∈L∞(\OT) the cost functional J(y)=∬\OTj1(t,x,u)+∫\Oj2(x,u(T,⋅)) where ∂tu−Δu=f(t,x,u)+y,u(0,⋅)=u0 with some classical boundary conditions, under constraints of the form −κ0≤y≤κ1 a.e.,∫\Oy(t,⋅)=V0. This class of problems arises in several application fields. A challenging feature of these problems is the study of the so-called abnormal set $ \{-\kappa_0
Subjects / Keywords
Reaction-diffusion equation; semi-linear parabolic equation; optimal control; second order optimality conditions; shape optimisation; two-scale expansions