An Improved Level Set Method for Reachability Problems in Differential Games

This study focuses on reachability problems in differential games. An improved level set method for computing reachable tubes is proposed in this paper. The reachable tube is described as a sublevel set of a value function, which is the viscosity solution of a Hamilton-Jacobi equation with running cost. We generalize the concept of reachable tubes and propose a new class of reachable tubes, which are referred to as cost-limited one. In particular, a performance index can be specified for the system, and a cost-limited reachable tube is a set of initial states of the system's trajectories that can reach the target set before the performance index increases to a given admissible cost. Such a reachable tube can be obtained by specifying the corresponding running cost function for the Hamilton-Jacobi equation. Different non-zero sublevel sets of the viscosity solution of the Hamilton-Jacobi equation at a certain time point can be used to characterize the cost-limited reachable tubes with different admissible costs (or the reachable tubes with different time horizons), thus reducing the storage space consumption. Several examples are provided to illustrate the validity and accuracy of the proposed method.