Singular arcs in the optimal control of a parabolic equation

International audienceWe present a theory of singular arc, and the corresponding second order necessary and sufficient conditions, for the optimal control of a semilinear parabolic equation with scalar control applied on the r.h.s. We obtain in particular an extension of Kelley's condition, and the characterization of a quadratic growth property for a weak norm.Nous présentons une théorie des arcs singuliers, et les conditions associées nécessaires ou suffisantes du second ordre, pour la commande optimale d'une équation parabolique avec commande scalaire appliquée au membre de droite.On obtient en particulier une extension de la condition de Kelley, et la caractérisation de la croissance quadratique dans une norme faible

State-constrained control-affine parabolic problems II: Second order sufficient optimality conditions

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 multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform