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

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 $y\in L^\infty(\OT)$ the cost functional $J(y)=\iint_\OT j_1(t,x,u)+\int_\O j_2(x,u(T,\cdot))$ where $\partial_t u-\Delta u=f(t,x,u)+y\,, u(0,\cdot)=u_0$ with some classical boundary conditions, under constraints of the form $-\kappa_0\leq y\leq \kappa_1\text{ a.e.}\,, \int_\O y(t,\cdot)=V_0$. 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

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. Localising optimality conditions for the linear optimal control of semilinear equations \emph{via} concentration results for oscillating solutions of linear parabolic equations. 2022. ⟨hal-03676505v2⟩