TY - GEN A1 - Hmede, Hiba A1 - Paquet, Luc A1 - Wachsmuth, Gerd T1 - No-gap second-order optimality conditions for additive manufacturing T2 - arXiv Y1 - 2024 UR - https://arxiv.org/abs/2410.18776 U6 - https://doi.org/10.48550/arXiv.2410.18776 ER - TY - GEN A1 - Hmede, Hiba A1 - Paquet, Luc A1 - Wachsmuth, Gerd T1 - No-gap second-order optimality conditions for additive manufacturing T2 - Mathematical control and related fields N2 - Additive manufacturing by laser fusion on a metal oxides powder bed has developed considerably in the last few years and allows to produce a wide range of complex parts. The mathematical models correspond to initial boundary value problems for the heat equation with moving heat sources according to the laser trajectories. The main questions concern the optimization of the trajectories scanned by the laser and of the thermal treatment time in order to melt the powder where it is desired to make the part and to minimize the thermal gradients. Our purpose in this current paper is to pursue the study of the optimization model that we have introduced in a previous paper. Here, we consider second-order optimality conditions for non-necessarily convex constraints on the laser paths. In particular, we obtain no gap between the second-order sufficient optimality condition and the necessary second-order optimality condition. To achieve this goal, we reformulate our optimal control problem in order to fit it in the framework of the abstract theory of optimization under constraints in Banach spaces. Higher regularity of the trajectories for local minimizers is also proved implying higher regularity of the corresponding Lagrange multipliers. The case of the regularity of the trajectories for stationary points is left open. KW - Optimal control problems KW - Parabolic equations KW - Heat equation with moving heat sources KW - Time of thermal treatment KW - Cost functionals KW - Abstract theory of optimization under constraints in Banach spaces KW - Higher regularity of the trajectories for local minimizers KW - Lagrange multipliers KW - Constraint qualifications KW - Second-order optimality conditions KW - Quadratic Legendre forms KW - Polyhedric sets. Y1 - 2025 U6 - https://doi.org/10.3934/mcrf.2025028 SN - 2156-8499 SP - 1 EP - 29 PB - AIMS CY - Springfield, MO ER -