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This paper presents Lax formulae for solving the following optimal controlproblems: minimize the maximum (or the minimum) cost over a time horizon, whilesatisfying a state constraint. We present a viscosity theory, and by applyingthe theory to the Hamilton-Jacobi (HJ) equations, these Lax formulae arederived. A numerical algorithm for the Lax formulae is presented: under certainconditions, this algorithm's computational complexity is polynomial in thedimension of the state. For each class of optimal control problem, an exampledemonstrates the use and performance of the Lax formulae.

Lax Formulae for Efficiently Solving Two Classes of State-Constrained Optimal Control Problems