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Radial and Non-Radial Solution Structures for Quasilinear Hamilton--Jacobi--Bellman Equations in Bounded Settings

Dragos-Patru Covei

TL;DR

This work proves existence and uniqueness of a positive classical solution $V$ to the quasilinear HJB equation $- rac{\sigma^{2}}{2}\Delta V + C_{\alpha}|\nabla V|^{p}-h(y)=0$ in a bounded smooth convex domain with Dirichlet data $V=g$, under sub-quadratic growth of $h$. The authors present a fully constructive proof via sub- and super-solution barriers, a comparison principle, and a monotone iteration scheme that yields a convergent sequence of linear Poisson problems; the approach is amenable to algorithmic implementation. They derive the PDE from stochastic optimal control with exit-time costs using the dynamic programming principle and Itô calculus, with the Legendre transform providing the explicit gradient term in the Hamiltonian. A radial symmetry result is established when the domain and data are symmetric, and the paper discusses computational strategies, including a practical numerical algorithm and software for both radial and nonradial settings. The findings have practical impact for applications in production planning and image restoration and offer avenues for future extensions to more general domains and higher regularity analyses.

Abstract

We study quasilinear Hamilton--Jacobi--Bellman equations on bounded smooth convex domains. We show that the quasilinear Hamilton--Jacobi--Bellman equations arise naturally from stochastic optimal control problems with exit-time costs. The PDE is obtained via dynamic programming applied to controlled Itô diffusions, providing both a probabilistic interpretation and a rigorous derivation. Our result establishes existence and uniqueness of positive classical solutions under sub-quadratic growth conditions on the source term. The constructive proofs, based on monotone iteration and barrier techniques, also provide a framework for algorithmic implementation with applications in production planning and image restoration. We provide complete detailed proofs with rigorous estimates and establish the connection to stochastic control theory through the dynamic programming principle.

Radial and Non-Radial Solution Structures for Quasilinear Hamilton--Jacobi--Bellman Equations in Bounded Settings

TL;DR

This work proves existence and uniqueness of a positive classical solution to the quasilinear HJB equation in a bounded smooth convex domain with Dirichlet data , under sub-quadratic growth of . The authors present a fully constructive proof via sub- and super-solution barriers, a comparison principle, and a monotone iteration scheme that yields a convergent sequence of linear Poisson problems; the approach is amenable to algorithmic implementation. They derive the PDE from stochastic optimal control with exit-time costs using the dynamic programming principle and Itô calculus, with the Legendre transform providing the explicit gradient term in the Hamiltonian. A radial symmetry result is established when the domain and data are symmetric, and the paper discusses computational strategies, including a practical numerical algorithm and software for both radial and nonradial settings. The findings have practical impact for applications in production planning and image restoration and offer avenues for future extensions to more general domains and higher regularity analyses.

Abstract

We study quasilinear Hamilton--Jacobi--Bellman equations on bounded smooth convex domains. We show that the quasilinear Hamilton--Jacobi--Bellman equations arise naturally from stochastic optimal control problems with exit-time costs. The PDE is obtained via dynamic programming applied to controlled Itô diffusions, providing both a probabilistic interpretation and a rigorous derivation. Our result establishes existence and uniqueness of positive classical solutions under sub-quadratic growth conditions on the source term. The constructive proofs, based on monotone iteration and barrier techniques, also provide a framework for algorithmic implementation with applications in production planning and image restoration. We provide complete detailed proofs with rigorous estimates and establish the connection to stochastic control theory through the dynamic programming principle.

Paper Structure

This paper contains 18 sections, 15 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Step 1: Construction of ordered positive sub- and super-solutions
  4. Step 2: Comparison principle
  5. Step 3: Existence by monotone iteration
  6. Step 4: Uniqueness
  7. Stochastic control derivation of the HJB equation
  8. The base model without regime switching (Lasry--Lions)
  9. Probability space, state and control
  10. Exit time and cost functional
  11. Dynamic programming and verification
  12. Legendre transform yielding the quasilinear elliptic PDE
  13. Radial Symmetry of Solutions
  14. Computational Implementation and Applications
  15. Numerical algorithm
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $h:\overline{\Omega }\rightarrow \lbrack 0,\infty )$ be continuous and sub-quadratic (i.e., there exist constants $C_{h}>0$ and $p:=\frac{\alpha }{\alpha -1}\in \lbrack 2,\infty )$ with $\alpha \in (1,2]$ such that $h(y)\leq C_{h}(1+|y|^{p})$ for all $y\in \overline{\Omega }$), and let $g\in \lb

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: Comparison Principle
  • proof
  • Lemma 2.2: First step monotonicity
  • proof
  • Lemma 2.3: Inductive monotonicity
  • proof
  • Corollary 2.4
  • proof
  • ...and 26 more