HAL CCSD
Existence of optimal shapes in parabolic bilinear optimal control problems
Mazari, Idriss
CEntre de REcherches en MAthématiques de la DEcision (CEREMADE) ; Université Paris Dauphine-PSL ; Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
Université Paris Dauphine-PSL ; Université Paris sciences et lettres (PSL)
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
https://hal.science/hal-04002274
2023
en
Bilinear optimal control
qualitative properties of optimisation problems
parabolic equations with measure datum
[MATH]Mathematics [math]
info:eu-repo/semantics/preprint
Preprints, Working Papers, ...
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
info:eu-repo/semantics/OpenAccess