Is Bellman Equation Enough for Learning Control?
This addresses a fundamental issue in value-based control methods for continuous spaces, which can converge to unstable solutions, impacting reinforcement learning and optimal control applications.
The paper tackles the problem of non-uniqueness in solutions to the Bellman equation for continuous state spaces in reinforcement learning, proving that linear systems admit at least $inom{2n}{n}$ solutions, with only one yielding an optimal and stable policy, and introduces a neural architecture to guarantee convergence to this stable solution.
The Bellman equation and its continuous-time counterpart, the Hamilton-Jacobi-Bellman (HJB) equation, serve as necessary conditions for optimality in reinforcement learning and optimal control. While the value function is known to be the unique solution to the Bellman equation in tabular settings, we demonstrate that this uniqueness fails to hold in continuous state spaces. Specifically, for linear dynamical systems, we prove the Bellman equation admits at least $\binom{2n}{n}$ solutions, where $n$ is the state dimension. Crucially, only one of these solutions yields both an optimal policy and a stable closed-loop system. We then demonstrate a common failure mode in value-based methods: convergence to unstable solutions due to the exponential imbalance between admissible and inadmissible solutions. Finally, we introduce a positive-definite neural architecture that guarantees convergence to the stable solution by construction to address this issue.