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

arXiv: https://arxiv.org/abs/2106.13440 · HTML: https://mlsanity.com/paper/2106.13440/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2021.

## Built on by (most cited first, 2 of 2)

- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [Hamilton-Jacobi Based Policy-Iteration via Deep Operator Learning](https://mlsanity.com/paper/2406.10920/) (2024)

## Abstract (arXiv)

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 textitthe 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.