Numerical solution of the HJB equation using a hybrid Bernstein–fractional Chebyshev spectral collocation method

Abstract

The Hamilton–Jacobi–Bellman (HJB) equation is fundamental in stochastic optimal control but is often difficult to solve accurately because of its nonlinear and multidimensional structure. This paper develops a hybrid tensor-product spectral collocation method based on shifted Bernstein polynomials in time and fractional-order Chebyshev polynomials in the state variables. The proposed shifted Bernstein–fractional Chebyshev (SBFC) framework allows the time and state directions to be approximated using different basis characteristics, while the fractional parameters provide adjustable resolution in the state domain. A weighted approximation framework is established, including density and convergence results for the mixed approximation space. The state collocation points are also selected according to the endpoint regularity of the fractional basis. The resulting HJB equation is reduced to a nonlinear algebraic system for the unknown expansion coefficients. Numerical experiments, including linear–quadratic control, resource extraction, and a two-dimensional stochastic control problem, show that SBFC provides accurate and competitive results compared with related spectral methods.