# On nouniqueness of solutions of Hamilton-Jacobi-Bellman equations

arXiv: https://arxiv.org/abs/1505.06567 · HTML: https://mlsanity.com/paper/1505.06567/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2015.

## Built on by (most cited first, 2 of 2)

- [Is Bellman Equation Enough for Learning Control?](https://mlsanity.com/paper/2503.02171/) (2025)
- [Optimality-Informed Neural Networks for Lunar Landing Trajectory Optimization](https://mlsanity.com/paper/2607.02741/) (2026)

## Abstract (arXiv)

An example of a nonunique solution of the Cauchy problem of Hamilton-Jacobi-Bellman (HJB) equation with surprisingly regular Hamiltonian is presented. The Hamiltonian H(t,x,p) is locally Lipschitz continuous with respect to all variables, convex in p and with linear growth with respect to p and x. The HJB equation possesses two distinct lower semicontinuous solutions with the same final conditions; moreover, one of them is the value function of the corresponding Bolza problem. The definition of lower semicontinuous solution was proposed by Barron-Jensen (1990) and Frankowska (1993). Using the example an analysis and comparison of assumptions in some uniqueness results in HJB equations is provided.