This paper presents a general Hamilton-Jacobi (HJ) framework for optimal control and two-player zero-sum game problems, both with state constraints. In the optimal control problem, a control signal and terminal time are determined to minimize the given cost and satisfy the state constraints. In the game problem, the two players interact via the system dynamics. Here, a strategy for each player, as well as a terminal time, are determined so that player A minimizes the cost and satisfies the state constraints while player B tries to prevent the success of player A. Dynamics, costs, and state...