State-constrained optimal control with the theory of functional connections: squared slacks and junction exponents

arXiv:2611.13047v1 Announce Type: new Abstract: For optimal control problems with an affine state constraint, we combine the theory of functional connections (TFC), which writes every function satisfying prescribed linear conditions as an explicit expression in terms of a free function, with the squared-slack substitution $b - a^\top x(t) = s(t)^2$. Every trial trajectory then satisfies the endpoint conditions and the path constraint identically. The accuracy depends on the regularity of the slack function at the junctions with a boundary arc. Along an optimal trajectory, the constraint residual vanishes like $(t_1 - t)^m$, where the junction exponent $m$ is equal to the order $p$ of the constraint if and only if the $p$-th derivative of the constraint jumps at the junction, and thus a single polynomial slack converges algebraically. The \emph{multiple-arc TFC method} that we introduce, with unknown junction times and with residuals $(t_1 - t)^m \sigma(t)^2$ on the free arcs, represents the optimal trajectory exactly and converges geometrically when the arcs are analytic. In a linear-quadratic class, an exact cost identity shows that the cost of every trial trajectory is an upper bound of the optimal cost. We compare both approaches with direct collocation and indirect shooting on the Bryson-Denham problem, on a rendezvous problem with an approach corridor and on the six-dimensional atmospheric reentry of a space shuttle with three state constraints.

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