On representation formulas for optimal control: A Lagrangian perspective

This paper studies the representation formulas for finite-horizon optimal control problems with or without state constraints, unifying two different viewpoints: the Lagrangian and dynamic programming frameworks. In a recent work by Lee and Tomlin [1], the generalised Lax formula is obtained via dynamic programming for optimal control problems with state constraints and non-linear systems. We revisit the formula from the Lagrangian perspective to provide a unified framework for understanding and implementing the non-trivial representation of the value function. Our simple derivation makes direct use of the Lagrangian formula from the theory of Hamilton–Jacobi 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 optimisation.