A maximum principle for a special optimal multiprocess problem with state constraints
Maria do Rosário de Pinho, Hasnaa Zidani · 2017
Here we prove a maximum principle for a special case of optimal multiprocess problems with state constraints. The class of problems of interest involves a family of three control systems each acting on three adjacent state regions, Ω1, Ω2and Ω3such that Ω1=∂Ω3=Ω2, where ∂Ωistands for boundary of the set Ωi. The aim is to drive the state from an initial position in the interior of Ω1to a target in the interior of Ω3. The three control systems are coupled by boundary constraints and the switching between control systems is dictated by state constraints. We improve on previous work (e.g., in [4]) since we cover the case when the state may or may not live in region Ω2and we allow some of data of the problem to be nonsmooth. Moreover, we propose an easy to verify condition to guarantee that the system can only be driven in Ω2by one and only one of the control systems. An interesting feature of our approach is that it can easily be generalized to cover more general cases, namely, with an odd number greater than 3 of regions.