OPTIMAL CONTROL OF THE TEMPERATURE BY THE LASER PATH AND THE THERMAL TREATMENT TIME IN SELECTIVE LASER MELTING PROCESS

Additive manufacturing by laser fusion on metal oxides powder beds such as, e.g., alumina (Al2O3) or aluminium titanate (Al2TiO5), has developed considerably in the last few years and today allows for the production of a wide range of complex objects. The mathematical problem considered is to control the temperature inside some part ohm of a powder layer. This phenomenon is governed by a parabolic initial boundary value problem with a heat source corresponding to the laser trajectory on some part of the boundary partial derivative ohm. The main questions concerning the optimization of the trajectories scanned by the laser on the boundary partial derivative ohm according to given criteria are as follows: imposing that during the thermal process the temperature reaches a melting value in the structure to be built, obtaining the desired temperature distribution at the end of the thermal process, minimizing the thermal gradients, and doing all this in the shortest possible thermal treatment time. To achieve this goal, we start by proving the existence of an optimal control, followed by first-order necessary optimality conditions. Finally, we establish a second-order sufficient optimality condition.