| Issue |
ESAIM: COCV
Volume 32, 2026
|
|
|---|---|---|
| Article Number | 5 | |
| Number of page(s) | 28 | |
| DOI | https://doi.org/10.1051/cocv/2025092 | |
| Published online | 21 January 2026 | |
Dynamic programming principle and Hamilton–Jacobi–Bellman equation for optimal control problems with uncertainty
1
Escola de Matemática Aplicada, FGV EMAp, Praia de Botafogo 190, 22250-900 Rio de Janeiro, RJ, Brazil
2
University of L’Aquila – Department of Information Engineering, Computer Science and Mathematics (DISIM), via Vetoio, 67100, L’Aquila, Italy
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
17
July
2024
Accepted:
23
November
2025
Abstract
We study the properties of the value function associated with an optimal control problem with uncertainties, known as average or Riemann–Stieltjes problem. Uncertainties are assumed to belong to a compact metric probability space, and appear in the dynamics, in the terminal cost and in the initial condition, which yield an infinite-dimensional formulation. By stating the problem as an evolution equation in a Hilbert space, we show that the value function is the unique lower semicontinuous proximal solution of the Hamilton–Jacobi–Bellman (HJB) equation. Our approach relies on invariance properties and the dynamic programming principle.
Mathematics Subject Classification: 35F21 / 49K45 / 49K27 / 49L25
Key words: Optimal control / Hamilton–Jacobi–Bellman equation / dynamic programming principle / invariance principles / Riemann–Stieltjes optimal control problems / uncertain dynamics
© The authors. Published by EDP Sciences, SMAI 2026
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