The generalization of Zermelo’s navigation problem with variable speed and limited acceleration

In this paper, we present a generalization of the Zermelo navigation problem considered by E. Zermelo. Our main objective in raising this problem is to find the optimal time of motion of a vehicle with variable speed and limited acceleration, in the presence of strong wind and moving or fixed obstacles. Contrary to other articles, in this paper, the moving speed is not fixed, and the limited acceleration is taken into account. The variable speed makes it possible to avoid collision with moving obstacles and therefore has an impact on the time-optimal trajectory. With the presence of obstacles, the problem involves continuous inequality constraints. To solve this problem, first of all, by parameterizing the problem and using the re-scaling technique, we obtain a set of parameters, such that our goal is to find the best choice of parameters to optimize the objective function. Here, the well-known penalty function is applied, and several examples are given to illustrate the performance of this method.