A Sobolev-Orthogonal Modal Collocation Method for Nonlinear Optimal Control Problems

This paper develops a Sobolev-orthogonal modal collocation method for nonlinear optimal control problems. The proposed transcription represents state and control trajectories by Sobolev-orthogonal polynomial expansions, with the corresponding modal coefficients serving as the decision variables of the nonlinear program. The governing dynamics are imposed through physical-time collocation residuals, while the objective functional is evaluated by positive Gauss–Legendre quadrature. By separating the Sobolev trial basis from the collocation grid, the formulation allows Chebyshev–Gauss–Lobatto, Legendre–Gauss–Lobatto, and Sobolev–Lobatto-type nodes to be examined within a unified modal framework. An approximation-level convergence analysis is established through Sobolev-orthogonal projection, best approximation in the Sobolev-induced norm, weighted derivative convergence, and physical-time residual consistency. Numerical experiments on hypersensitive, moon-lander, and fixed-final-time Goddard rocket benchmarks demonstrate that the method can approximate optimal trajectories involving endpoint boundary layers, bang-bang controls, and singular-thrust structures. The results further indicate that polynomial resolution primarily governs objective and terminal-state accuracy, whereas node distribution and the Sobolev parameter influence trajectory reconstruction, conditioning, and finite-precision behavior. These findings suggest that Sobolev-orthogonal modal collocation provides a derivative-aware direct discretization framework for nonlinear optimal control.

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