  We consider a reach-avoid differential game, in which one of the players aims
to steer the system into a target set without violating a set of state
constraints, while the other player tries to prevent the first from succeeding;
the system dynamics, target set, and state constraints may all be time-varying.
The analysis of this problem plays an important role in collision 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, targets and constraints without the need for state augmentation, we
propose a modified Hamilton-Jacobi-Isaacs equation in the form of a
double-obstacle variational inequality, and prove that the zero sublevel set of
its viscosity solution 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 respect to the time-invariant case. We provide an
implementation of this method based on well-known numerical schemes and show
its convergence through a simple example; we include a second example in which
our method substantially outperforms the state augmentation approach.
