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On classical solutions and canonical transformations for Hamilton--Jacobi--Bellman equations

Mohit Bansil, Alpár R. Mészáros

TL;DR

The paper addresses global classical solvability of Hamilton--Jacobi--Bellman equations beyond the fully convex regime by leveraging a family of upper-triangular canonical transformations $ (x,p)\mapsto (x,p-\alpha x) $. It shows that the HJB is invariant under this transform and that appropriate $\alpha$ can convexify transformed data $H_\alpha$ and $G_\alpha$, yielding global $C^{1,1}_{loc}$ solvability for the original problem. Through complementary Lagrangian and Hamiltonian analyses, it derives explicit, verifiable derivative-based conditions (in terms of $\lambda_0$, $\lambda_H$, $\lambda_G$) that guarantee global well-posedness when $\alpha$ is large enough. This approach provides a new mechanism to extend global well-posedness results beyond the fully convex setting and suggests potential extensions to nonlinear transforms and mean-field game frameworks.

Abstract

In this note we show how canonical transformations reveal hidden convexity properties for deterministic optimal control problems, which in turn result in global existence of $C^{1,1}_{loc}$ solutions to first order Hamilton--Jacobi--Bellman equations.

On classical solutions and canonical transformations for Hamilton--Jacobi--Bellman equations

TL;DR

The paper addresses global classical solvability of Hamilton--Jacobi--Bellman equations beyond the fully convex regime by leveraging a family of upper-triangular canonical transformations . It shows that the HJB is invariant under this transform and that appropriate can convexify transformed data and , yielding global solvability for the original problem. Through complementary Lagrangian and Hamiltonian analyses, it derives explicit, verifiable derivative-based conditions (in terms of , , ) that guarantee global well-posedness when is large enough. This approach provides a new mechanism to extend global well-posedness results beyond the fully convex setting and suggests potential extensions to nonlinear transforms and mean-field game frameworks.

Abstract

In this note we show how canonical transformations reveal hidden convexity properties for deterministic optimal control problems, which in turn result in global existence of solutions to first order Hamilton--Jacobi--Bellman equations.
Paper Structure (3 sections, 6 theorems, 39 equations)

This paper contains 3 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Canonical transformations and classical solutions from the Lagrangian perspective
  3. Canonical transformations and classical solutions from the perspective of Hamiltonian systems

Key Result

Theorem 1.1

Let $\alpha\in\mathbb{R}$. Then $u$ is a classical solution to eq:HJ with data $(H,G)$ in $(0,T)\times\mathbb{R}^d$, if and only if $u_\alpha:(0,T)\times\mathbb{R}^{d}$, defined as is a classical solution to eq:HJ on $(0,T)\times\mathbb{R}^d$ with data $(H_\alpha,G_\alpha).$

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • proof : Proof of Theorem \ref{['prop:trans_HJ']}
  • Remark 2.2
  • Lemma 3.1
  • ...and 6 more