  We present novel results on the solution of a class of leavable, undiscounted
optimal control problems in the minimax sense for nonlinear, continuous-state,
discrete-time plants. The problem class includes entry-(exit-)time problems as
well as minimum time, pursuit-evasion and reach-avoid games as special cases.
We utilize auxiliary optimal control problems (`abstractions') to compute both
upper bounds of the value function, i.e., of the achievable closed-loop
performance, and symbolic feedback controllers realizing those bounds. The
abstractions are obtained from discretizing the problem data, and we prove that
the computed bounds and the performance of the symbolic controllers converge to
the value function as the discretization parameters approach zero. In
particular, if the optimal control problem is solvable on some compact subset
of the state space, and if the discretization parameters are sufficiently
small, then we obtain a symbolic feedback controller solving the problem on
that subset. These results do not assume the continuity of the value function
or any problem data, and they fully apply in the presence of hard state and
control constraints.
