An Improved Level Set Method for Reachability Problems in Differential Games

This study focuses on reachability problems in differential games. An improved level set (LS) method for computing reachable tubes (RTs) is proposed in this article. The RT is described as a sub-LS of a value function, which is the viscosity solution of a Hamilton–Jacobi (HJ) equation with running cost. We generalize the concept of RTs and propose a new class of RTs, which are referred to as cost-limited one. In particular, a performance index can be specified for the system, and A set of initial states of the system's evolutions that can reach the target set before the performance index grows to a given allowable cost is referred to as a cost-limited RT (CRT). Such an RT can be obtained by specifying the corresponding running cost function for the HJ equation. Different nonzero sub-LSs of the viscosity solution of the HJ equation at a certain time point can be used to characterize the CRTs with different allowable costs (or the RTs with different time horizons), thus reducing the storage space consumption. The validity and accuracy of the suggested technique are demonstrated via some examples.