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

A Computationally Efficient Hamilton-Jacobi-based Formula for State-Constrained Optimal Control Problems