  This paper presents Hamilton-Jacobi (HJ) formulations for two classes of
two-player zero-sum games: one with a maximum cost value over time, and one
with a minimum cost value over time. In the zero-sum game setting, player A
minimizes the given cost while satisfying state constraints, and player B wants
to prevent player A's success. For each class of problems, this paper presents
two HJ equations: one for time-varying dynamics, cost, and state constraint;
the other for time-invariant dynamics, cost, and state constraint. Utilizing
the HJ equations, the optimal control for each player is analyzed, and a
numerical algorithm is presented to compute the solution to the HJ equations. A
two-dimensional water system is introduced as an example to demonstrate the
proposed HJ framework.
