A reach-avoid game is one in which an agent attempts to reach a predefined goal, while avoiding some adversarial circumstance induced by an opposing agent or disturbance. Their analysis plays an important role in problems such as safe motion planning and obstacle avoidance, yet computing solutions is often computationally expensive due to the need to consider adversarial inputs. In this work, we present an open-loop formulation of a two-player reach-avoid game whereby the players define their control inputs prior to the start of the game. We define two open-loop games, each of which is conservative towards one player, show how the solutions to these games are related to the optimal feedback strategy for the closed-loop game, and demonstrate a modified Fast Marching Method to efficiently compute those solutions.
Abstract. In this work we provide a novel approach to homogenization for a class of static Hamilton-Jacobi (HJ) equations, which we call metric HJ equations. We relate the solutions of the HJ equations to the distance function in a corresponding Riemannian or Finslerian metric. The metric approach allows us to conclude that the homogenized equation also induces a metric. The advantage of the method is that we can solve just one auxiliary equation to recover the homogenized HamiltonianH(p). This is significant improvement over existing methods which require the solution of the cell problem (or a variational problem) for each value of p. Computational results are presented and compared with analytic results when available for piece-wise constant periodic and random speed functions.
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