hal-04002274
https://hal.science/hal-04002274
https://hal.science/hal-04002274/document
https://hal.science/hal-04002274/file/M-2023-FP-2%20%281%29.pdf
[CNRS] CNRS - Centre national de la recherche scientifique
[UNIV-DAUPHINE] Université Paris Dauphine - Paris IX
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[CEREMADE] CEntre de REcherches en MAthématiques de la DEcision
[PSL] Université Paris sciences et lettres
[UNIV-DAUPHINE-PSL] Université Paris Dauphine - PSL
Existence of optimal shapes in parabolic bilinear optimal control problems
Mazari, Idriss
[MATH] Mathematics [math]
UNDEFINED
Bilinear optimal control
qualitative properties of optimisation problems
parabolic equations with measure datum
The aim of this paper is to prove the existence of optimal shapes in bilinear parabolic optimal control problems. We consider a parabolic equation that writes ∂tum − ∆um = f (t, x, um) + mum. The set of admissible controls is given by A = {m ∈ L ∞ , m− m m+ a.e., ´Ω m(t, •) = V1(t)} where m± = m±(t, x) are two reference functions in L ∞ ((0, T) × Ω), and where V1 = V1(t) is a reference integral constraint. The functional to optimise is J : m → ˜j1(um) + ´Ω j2(um(T)). Roughly speaking we prove that, if j1 and j2 are non-decreasing and if one is increasing any solution of maxA J is bang-bang: any optimal m * writes m * = 1Em− + 1Ec m+ for some E ⊂ (0, T) × Ω. From the point of view of shape optimization, this is a parabolic analog of the Buttazzo-Dal Maso theorem in shape optimisation. The proof is based on second-order criteria and on an approximation-localisation procedure for admissible perturbations. This last part uses the theory of parabolic equations with measure datum.
2023-02-23
2023-02-23
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