# On representation formulas for optimal control: A Lagrangian perspective

arXiv: https://arxiv.org/abs/2204.02050 · HTML: https://mlsanity.com/paper/2204.02050/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2022.

## Built on by (most cited first, 1 of 1)

- [Hamilton-Jacobi Based Policy-Iteration via Deep Operator Learning](https://mlsanity.com/paper/2406.10920/) (2024)

## Abstract (arXiv)

In this paper, we study representation formulas for finite-horizon optimal control problems with or without state constraints, unifying two different viewpoints: the Lagrangian and dynamic programming (DP) frameworks. In a recent work [1], the generalized Lax formula is obtained via DP for optimal control problems with state constraints and nonlinear systems. We revisit the formula from the Lagrangian perspective to provide a unified framework for understanding and implementing the nontrivial representation of the value function. Our simple derivation makes direct use of the Lagrangian formula from the theory of Hamilton-Jacobi (HJ) equations. We also discuss a rigorous way to construct an optimal control using a δ-net, as well as a numerical scheme for controller synthesis via convex optimization.