Abstract

In this article, we derive a maximum principle for a special class of free end time optimal control of multiprocesses involving a family of control systems acting in different regions defined by state constraints. We are mainly interested in problems with contiguous time intervals. The main feature of our maximum principle is that it covers the case where some of the regions considered may not be visited. This means that the intervals where the corresponding control systems are active may be reduced to a single point. The derivation of our maximum principle is done by reformulating the optimal multiprocess problem as an equivalent fixed time state constrained optimal control problem. This reformulated problem is also of interest since it provides the means to solve optimal multiprocess problems numerically via the direct method. We illustrate our findings with an example concerning the path planning of an autonomous underwater vehicle (AUV) using a simple kinematic model derived for simulations. We use simplified point-mass model for the motion of an AUV in a horizontal plane and we assume that the ocean currents are known. We recast this problem as the multiprocess optimal control problem of interest and we study it via simulations presenting computational results partially validated by the maximum principle.