SOLUTION THEORY OF HAMILTON--JACOBI--BELLMAN EQUATIONS IN SPECTRAL BARRON SPACES

We study the solution theory of the whole-space static (elliptic) Hamilton--Jacobi--Bellman (HJB) equation in spectral Barron spaces. We prove that under the assumption that the coefficients involved are spectral Barron functions and the discount factor is sufficiently large, there exists a sequence of uniformly bounded spectral Barron functions that converges locally uniformly to the solution. As a consequence, the solution of the HJB equation can be approximated by two-layer neural networks without curse of dimensionality.