Error analysis for pod approximations of infinite horizon problems via the dynamic programming approach

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.