Collisions are common in many dynamical systems with real applications. They
can be formulated as hybrid dynamical systems with discontinuities
automatically triggered when states transverse certain manifolds. We present an
algorithm for the optimal control problem of such hybrid dynamical systems,
based on solving the equations derived from the hybrid minimum principle (HMP).
The algorithm is an iterative scheme following the spirit of the method of
successive approximations, and it is robust to undesired collisions observed in
the initial guesses. We carefully analyze and address several numerical
challenges introduced by the discontinuities. The algorithm is tested on disc
collision problems whose optimal solutions exhibit one or multiple collisions.
Linear convergence in terms of the iteration steps and asymptotic first-order
accuracy in terms of time discretization are observed when the algorithm is
implemented with the forward-Euler scheme. The numerical results demonstrate
that the proposed algorithm has better accuracy and convergence than direct
methods based on gradient descent. The algorithm is also simpler, more
accurate, and more stable than a deep reinforcement learning method.

提出一种解决混合动力系统最优控制问题的算法，通过解决混合最小原则 (HMP) 导出的方程，采用逐步逼近法，解决了由不连续性产生的多个数字挑战，其准确性和收敛性均优于基于梯度下降的直接方法和深度强化学习方法。