  Weighted timed games are zero-sum games played by two players on a timed
automaton equipped with weights, where one player wants to minimise the
cumulative weight while reaching a target. Used in a reactive synthesis
perspective, this quantitative extension of timed games allows one to measure
the quality of controllers in real-time systems. Weighted timed games are
notoriously difficult and quickly undecidable, even when restricted to
non-negative weights. For non-negative weights, the largest class that can be
analysed has been introduced by Bouyer, Jaziri and Markey in 2015. Though the
value problem is undecidable, the authors show how to approximate the value by
considering regions with a refined granularity. In this work, we extend this
class to incorporate negative weights, allowing one to model energy for
instance, and prove that the value can still be approximated, with the same
complexity. A small restriction also allows us to obtain a class of decidable
weighted timed games with negative weights and an arbitrary number of clocks.
In addition, we show that a symbolic algorithm, relying on the paradigm of
value iteration, can be used as an approximation/computation schema over these
classes. We also consider the special case of untimed weighted games, where the
same fragments are solvable in polynomial time: this contrasts with the
pseudo-polynomial complexity, known so far, for weighted games without
restrictions.
