Optimality conditions for optimal control problems with respect to the initial condition via a Laplace type method and two-scales like expansions

Idriss Mazari (1, 2) , Grégoire Nadin (3, 4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Université Paris Dauphine-PSL
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 CNRS - Centre National de la Recherche Scientifique

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 u0 ∈ L ∞ (Ω) the cost functional J(u0) = ˜(0;T)×Ω j1(t, x, u) + ´Ω j2(x, u(T, •)) where ∂tu − ∆u = f (t, x, u) , u(0, •) = u0 with some classical boundary conditions, under constraints of the form −κ0 ≤ u0 ≤ κ1 a.e. , ´Ω u0 = 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 {−κ0 < u * 0 < κ1} where u * 0 is an optimiser. This set is in general non-empty and it is important (for instance for numerical applications) to understand the behaviour of u * 0 in this set: which values can u * 0 take? In this paper, we introduce a Laplace-type method to provide some answers to this question. This Laplace type method is of independent interest.

Mots clés: Reaction-diffusion equation; semi-linear parabolic equation; optimal control; second order optimality conditions; shape optimisation; two-scale expansions

Domaines: Optimisation et contrôle [math.OC]

Idriss Mazari, Grégoire Nadin. Optimality conditions for optimal control problems with respect to the initial condition via a Laplace type method and two-scales like expansions. 2022. ⟨hal-03676505v1⟩