Relaxation theorems and necessary optimality conditions for semiconvex multidimensional control problems

迄今为止，多维控制问题的庞特里亚金极大原理的证明与被积域和控制域的凸性或凸松弛性密切相关。为了克服这一限制，该提案旨在推导一阶PDE多维控制问题的一阶最优性条件，该问题从一开始就涉及多凸或拟凸数据或需要非凸松弛。证明技术，它结合了多凸分析的元素与离散化方法和梯度杨测度的性质，将被应用。必要最优性条件的证明与半向量松弛之间的联系，对于“向量”情况下的问题是必需的，将被系统地研究。

Up to now, the proof of Pontryagin's maximum principle for multidimensional control problems is closely related to the convexity or the convex relaxability of integrand and control domain. In order to overcome this limitation, the proposal aims to derive first-order optimality conditions for multidimensional control problems with first-order PDE's involving either polyconvex or quasiconvex data from the outset or requiring nonconvex relaxation. Proof techniques, which combine elements of polyconvex analysis with discretization methods and properties of gradient Young measures, will be applied. The connections between the proof of necessary optimality conditions and the semiconvex relaxation, which is mandatory for problems in the "vectorial" case, shall be systematically investigated.