In this paper infinite horizon optimal control problems for nonlinear high-dimensional dynamical systems are studied. Nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation which stems from the dynamic programming approach. However, the bottleneck is mainly due to the curse of dimensionality, and HJB equations are solvable only in a relatively small dimension. Therefore, a reduced-order model is derived for the dynamical system, using the method of proper orthogonal decomposition (POD). The resulting errors in the HJB equations are estimated by an a priori error analysis, which is utilized in the numerical approximation to ensure a desired accuracy for the POD method. Numerical experiments illustrates the theoretical findings.