Time-optimal control of a self-propelled particle in a spatiotemporal flow field

We address a minimum-time problem that constitutes an extension of the classical Zermelo navigation problem in higher dimensions. In particular, we address the problem of steering a self-propelled particle to a prescribed terminal position with free terminal velocity in the presence of a spatiotemporal flow field. Furthermore, we assume that the norm of the rate of change of the particle's velocity relative to the flow is upper bounded by an explicit upper bound. To address the problem, we first employ Pontryagin's minimum principle to parameterise the set of candidate time-optimal control laws in terms of a parameter vector that belongs to a compact set. Subsequently, we develop a simple numerical algorithm for the computation of the minimum time-to-come function that is tailored to the particular parametrisation of the set of the candidate time-optimal control laws of our problem. The proposed approach bypasses the task of converting the optimal control problem to a parameter optimisation problem, which can be computationally intense, especially when one is interested in characterising the optimal synthesis of the minimum-time problem. Numerical simulations that illustrate the theoretical developments are presented.