Convex Pursuit-Evasion Games for Spacecraft Proximity Operations on Circular and Elliptical Orbits

Abstract

Inspecting or servicing a non-cooperative spacecraft is a two-player game: the target can thrust to defeat the inspector's plan. Hamilton-Jacobi-Isaacs reachability answers such games exactly but its cost grows exponentially with state dimension, while learning-based controllers scale yet certify nothing. Convex formulations are the usual escape, typically cast as convex-concave saddle-point problems. That framing fails in the terminal-distance orbital game: with effort regularization, the payoff is convex in both controls, so no pure open-loop saddle point need exist. In its place we derive two exact one-sided guarantees read from the players' terminal reachable sets via support functions: an escape certificate proving the target can hold a guaranteed standoff, and a security strategy bounding the miss distance the inspector can force. Together they bracket the engagement without assuming player rationality, and over two hundred perturbed trials the escape certificate separated capture from escape without error. A projected extragradient method supplies a strategy pair in about 25 ms, certified in place by best-response gaps. The construction carries unchanged to elliptical reference orbits via Yamanaka-Ankersen dynamics, where the orbital phase moves miss distance by nearly a factor of two, reproduced by nonlinear Keplerian propagation. Under receding-horizon play the inspector captures in half of ten representative engagements. Angles-only navigation error, keep-out zones, multiple pursuers, and closed-loop play are treated as extensions.