  Presented is a new method for calculating the time-optimal guidance control
for a multiple vehicle pursuit-evasion system. A joint differential game of k
pursuing vehicles relative to the evader is constructed, and a
Hamilton-Jacobi-Isaacs (HJI) equation that describes the evolution of the value
function is formulated. The value function is built such that the terminal cost
is the squared distance from the boundary of the terminal surface.
Additionally, all vehicles are assumed to have bounded controls. Typically, a
joint state space constructed in this way would have too large a dimension to
be solved with existing grid-based approaches. The value function is computed
efficiently in high-dimensional space, without a discrete grid, using the
generalized Hopf formula. The optimal time-to-reach is iteratively solved, and
the optimal control is inferred from the gradient of the value function.
