Thomas A. Henzinger, Benjamin Horowitz, and Rupak Majumdar
In order to study control problems for hybrid systems, we generalize hybrid automata to hybrid games --say, controller vs. plant. If we specify the continuous dynamics by constant lower and upper bounds, we obtain rectangular games. We show that for rectangular games with objectives expressed in LTL (linear temporal logic), the winning states for each player can be computed, and winning strategies can be synthesized. Our result is sharp, as already reachability is undecidable for generalizations of rectangular systems, and optimal --singly exponential in the size of the game structure and doubly exponential in the size of the LTL objective. Our proof systematically generalizes the theory of hybrid systems from automata (single-player structures) to games (multi-player structures): we show that the successively more general infinite-state classes of timed, 2D rectangular, and rectangular games induce successively weaker, but still finite, quotient structures called game bisimilarity, game similarity, and game trace equivalence. These quotients can be used, in particular, to solve the LTL control problem.
Proceedings of the Tenth International Conference on Concurrency Theory (CONCUR), Lecture Notes in Computer Science 1664, Springer, 1999, pp. 320-335.