On representation formulas for optimal control: A Lagrangian perspective

TL;DR

This paper unifies Lagrangian and dynamic programming approaches to derive representation formulas for finite-horizon optimal control problems, facilitating better understanding and numerical implementation.

Contribution

It provides a Lagrangian perspective on the generalized Lax formula, connecting Hamilton-Jacobi theory with optimal control representations.

Findings

Unified framework for Lagrangian and DP viewpoints

A simple derivation using Hamilton-Jacobi equations

Numerical scheme for controller synthesis via convex optimization

Abstract

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 $\delta$-net, as well as a numerical scheme for controller synthesis via convex optimization.