Jacobi Integral Pseudospectral Method for Solving Infinite-Horizon Optimal Control Problems

We introduce a novel direct integral pseudospectral (IPS) method for addressing a class of infinite-horizon optimal control problems (IHOCs) with continuous time. This approach transforms IHOCs into finite-horizon optimal control problems (FHOCs) in integral form through specific parametric mappings, which are then discretized into finite-dimensional nonlinear programming problems (NLPs) using rational collocations based on Jacobi polynomials and Jacobi-Gauss-Radau (JGR) nodes. Our method extends previous work that utilized Gegenbauer polynomials by employing the more general and flexible Jacobi polynomial family, which offers additional degrees of freedom through its two parameters $\alpha$ and $\beta$. We provide a comprehensive analysis of the interplay between parametric mappings, barycentric rational collocations based on Jacobi polynomials and JGR points, and the convergence properties of the collocated solutions. The paper presents a rigorous examination of the method's error bounds and convergence characteristics, along with a stability analysis based on the Lebesgue constant for JGR-based rational interpolation. We validate our theoretical findings through two illustrative examples, including a practical application to spacecraft attitude maneuvers. Our results demonstrate that the proposed collocation method, when combined with an efficient NLP solver (MATLAB's fmincon solver), converges exponentially to near-optimal approximations for coarse collocation mesh grid sizes. Furthermore, we show that certain parameter combinations with $\alpha \neq \beta$ yield more accurate solutions than those achievable with Gegenbauer, Legendre, or Chebyshev polynomials.