Conjecture on divergence of classical Jacobi polynomial collocations

Let $T_{1,L}^{(\alpha)}$ and $T_{2,L}^{(\alpha)}$ be the maps used to transform the infinite-horizon optimal control problem, and consider classical Jacobi polynomial collocations of the FHOC in differential or integral form using either map. Assume that the computations use floating-point arithmetic and that the discretization uses a single mesh grid, consisting of either Gauss/Gauss-Radau nodes or equally spaced nodes.

Divergence conjecture. Such collocations will likely diverge as the mesh size grows large. For Gauss/Gauss-Radau discretizations, the divergence is attributed to the divergence established in the present analysis; for equally spaced discretizations, it is attributed to Runge's phenomenon and the ill-conditioning of polynomial interpolation as the polynomial degree grows.

The conjecture concerns numerical stability of single-grid classical Jacobi collocations for infinite-horizon optimal control problems. The Gauss/Gauss-Radau case is motivated by the paper's divergence analysis, whereas the equally spaced case is expected to fail because of interpolation instability.