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

An Improved Level Set Method for Reachability Problems in Differential Games