A Maximum Principle for a Special Optimal Multiprocess Problem with State Constraints

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, Omega(1), Omega(2) and Omega(3) such that partial derivative Omega(1) - partial derivative Omega(3) - partial derivative Omega(2), where partial derivative Omega(i) stands for boundary of the set Omega(i). The aim is to drive the state from an initial position in the interior of Omega(1) to a target in the interior of Omega(3). 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 Omega(2) and 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 Omega(2) by 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.