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In this talk, we address optimal control problems with spatially heterogeneous dynamics, where the state space is partitioned into regions and the system is governed by a different smooth vector field in each region. We first investigate the derivation of a hybrid maximum principle through an augmentation technique. A counterexample shows that the augmented solution need not be a local minimizer in the usual sense. This motivates the introduction of a suitable notion of local optimality and a corresponding Pontryagin maximum principle, from which we derive the hybrid maximum principle. As an application, we consider optimal control problems with loss-control regions, where the control must remain constant while the trajectory evolves within certain regions. We present necessary optimality conditions and a two-step numerical approach: a direct method applied to a regularized problem, followed by an indirect shooting method for the original problem. Finally, we introduce a regularization approach for optimal control problems on stratified domains based on additional controls. Under suitable assumptions, we establish that a subsequence of solutions to the regularized problems converges to a solution of the original problem. Numerical simulations illustrate these methods.

Biography: Anas Bouali is a Research and Teaching Fellow at the Institut de Mathématiques de Toulouse (France). He obtained his PhD in Mathematics from Avignon University (France) in 2023 and subsequently held a two-year postdoctoral position at the French National Research Institute for Agriculture, Food and Environment in Montpellier (France). His research focuses on the theoretical and numerical analysis of optimal control problems involving hybrid and nonsmooth dynamical systems, including optimality conditions, regularization methods, and applications to epidemiology.