Weighted timed games (WTGs for short) are two-player zero-sum games played in a timed automaton equipped with integer weights into transitions and locations. In a turn-based fashion, the current player chooses the next delay and transition. We consider optimal reachability objectives, in which one of the players, whom we call Min, wants to reach a target location while minimising the cumulated weight given by the sequence of transitions firing and the time spent in each location. Its opponent, called Max, has the opposite objective, i.e. avoids the target or maximises the accumulated weight needed to reach the target.

This allows one to define the value of a WTG as the minimal weight Min can guarantee whatever Max does. In a WTG, the value for a given location is a function according to the valuation of the clocks when the first player starts to play.

While knowing if Min has a strategy to guarantee the value lower than a given threshold is known to be undecidable (with two or more clocks), several conditions, one of them being the divergence or one-clock WTGs have been given to recover decidability. Decidability proof relies on the properties of the value: it is a piecewise affine function characterise as a fixpoint of a operator that locally chooses the best option for each player.

I propose to study this function for all WTGs: for which class of WTGs, the value function is continuous and/or a fixed point of the local operator? In particular, I propose to start with WTGs that only use non-negative weights.