Control problems for embedded systems like cars and trains can be modeled by two-player hybrid games. Control envelopes, which are families of safe control solutions, correspond to nondeterministic policies that ensure a player following them will not lose. Each deterministic, finite specialization of the nondeterministic policy is a control solution. This paper synthesizes control envelopes for hybrid games that are as permissive as possible. It introduces subvalue maps , a compositional representation of such policies that enables verification and synthesis along the structure of the game. An inductive logical characterization in differential game logic (dGL) checks whether a subvalue map induces a sound control envelope which ensures that the player never loses, no matter what actions the opponent plays. The maximal subvalue map, which allows the most action options while still winning, is shown to exist and satisfy a logical characterization. An inductive subvalue map synthesis framework is obtained from the soundness characterization. An evaluation of the framework uses the significant expressivity of dGL to model and solve a broad range of control challenges.

@article{paperbot3804,
  title = {Hybrid Game Control Envelope Synthesis},
  author = {Aditi Kabra and Jonathan Laurent and Stefan Mitsch and André Platzer},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {10},
  number = {OOPSLA1},
  year = {2026},
  doi = {10.1145/3798230}
}