  This paper investigates a Hamilton-Jacobi (HJ) analysis to solve
finite-horizon optimal control problems for high-dimensional systems. Although
grid-based methods, such as the level-set method [1], numerically solve a
general class of HJ partial differential equations, the computational
complexity is exponential in the dimension of the continuous state. To manage
this computational complexity, methods based on Lax-Hopf theory have been
developed for the state-unconstrained optimal control problem under certain
assumptions, such as affine dynamics and state-independent stage cost. Based on
the Lax formula [2], this paper proposes an HJ formula for the
state-constrained optimal control problem for nonlinear systems. We call this
formula \textit{the generalized Lax formula} for the optimal control problem.
The HJ formula provides both the optimal cost and an optimal control signal. We
also provide an efficient computational method for a class of problems for
which the dynamics is affine in the state, and for which the stage and terminal
cost, as well as the state constraints, are convex in the state. This class of
problems does not require affine dynamics and convex stage cost in the control.
This paper also provides three practical examples.
