Reach-Avoid Problems with Time-Varying Dynamics, Targets and Constraints

We consider a reach-avoid differential game, in which one of the players aims to steer the system into a target set with-out violating a set of state constraints, while the other player tries to prevent the first from succeeding; the system dynam-ics, target set, and state constraints may all be time-varying. The analysis of this problem plays an important role in colli-sion avoidance, motion planning and aircraft control, among other applications. Previous methods for computing the guaranteed winning initial conditions and strategies for each player have either required augmenting the state vector to include time, or have been limited to problems with either no state constraints or entirely static targets, constraints and dynamics. To incorporate time-varying dynamics, tar-gets and constraints without the need for state augmenta-tion, we propose a modified Hamilton-Jacobi-Isaacs equa-tion in the form of a double-obstacle variational inequality, and prove that the zero sublevel set of its viscosity solu-tion characterizes the capture basin for the target under the state constraints. Through this formulation, our method can compute the capture basin and winning strategies for time-varying games at no additional computational cost with re-spect to the time-invariant case. We provide an implementa-tion of this method based on well-known numerical schemes and show its convergence through a simple example; we in-clude a second example in which our method substantially outperforms the state augmentation approach. 1