Stable representations of Hamilton-Jacobi-Bellman equations with infinite horizon

TL;DR

This work addresses the infinite-horizon state-constrained HJB equation by recasting the Hamiltonian as an epigraphical representation suitable for the Basco-Frankowska framework. It develops a representation $(IB,f,l)$ of a convex $H(t,x,p)$ so that $H(t,x,p)$ equals a supremum over $IB$ of an affine form in $p$, with $f(t,x,IB)=dom(H^*(t,x,·))$, enabling existence/uniqueness results for weak solutions via the calculus-of-variations formulation. The paper proves stability of the representations under perturbations and establishes convergence of the value functions via an $e$-liminf framework, supported by forward viability and backward invariance conditions. Through explicit examples, it clarifies when weaker conditions suffice and highlights the relationship between $(B)$ and $(B)_H$ in the epigraphical setting.

Abstract

In this paper, for the Hamilton-Jacobi-Bellman equation with an infinite horizon and state constraints, we construct a suitably regular representation. This allows us to reduce the problem of existence and uniqueness of solutions to the Frankowska and Basco theorem from (2019). Furthermore, we demonstrate that our representations are stable. The obtained results are illustrated with examples.

Theorems & Definitions (31)

• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Corollary 3.4
• Theorem 3.5
• Theorem 3.6
• Remark 3.7
• Example 3.8
• Example 3.9
• Example 3.10