  This paper presents Lax formulae for solving the following optimal control
problems: minimize the maximum (or the minimum) cost over a time horizon, while
satisfying a state constraint. We present a viscosity theory, and by applying
the theory to the Hamilton-Jacobi (HJ) equations, these Lax formulae are
derived. A numerical algorithm for the Lax formulae is presented: under certain
conditions, this algorithm's computational complexity is polynomial in the
dimension of the state. For each class of optimal control problem, an example
demonstrates the use and performance of the Lax formulae.
