State-constrained control-affine parabolic problems I: First and Second order necessary optimality conditions

M Soledad Aronna (1) , J. Frederic Bonnans (2) , Axel Kröner (3)

1 FGV/EMAp - Fundação Getulio Vargas - Escola de Matemática Aplicada [Rio de Janeiro]
2 COMMANDS - Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires
3 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimen-sional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results.

Domaines: Optimisation et contrôle [math.OC]

M Soledad Aronna, J. Frederic Bonnans, Axel Kröner. State-constrained control-affine parabolic problems I: First and Second order necessary optimality conditions. Set-Valued and Variational Analysis, inPress. ⟨hal-03011004⟩