  Weighted Timed Games (WTG for short) are the most widely used model to
describe controller synthesis problems involving real-time issues. The
synthesized strategies rely on a perfect measure of time elapse, which is not
realistic in practice. In order to produce strategies tolerant to timing
imprecisions, we rely on a notion of robustness first introduced for timed
automata. More precisely, WTGs are two-player zero-sum games played in a timed
automaton equipped with integer weights in which one of the players, that we
call Min, wants to reach a target location while minimising the cumulated
weight. In this work, we equip the underlying timed automaton with a semantics
depending on some parameter (representing the maximal possible perturbation) in
which the opponent of Min can in addition perturb delays chosen by Min.
  The robust value problem can then be stated as follows: given some threshold,
determine whether there exists a positive perturbation and a strategy for Min
ensuring to reach the target, with an accumulated weight below the threshold,
whatever the opponent does.
  We provide the first decidability result for this robust value problem by
computing the robust value function, in a parametric way, for the class of
divergent WTGs (introduced to obtain decidability of the (classical) value
problem in WTGs without bounding the number of clocks). To this end, we show
that the robust value is the fixpoint of some operators, as is classically done
for value iteration algorithms. We then combine in a very careful way two
representations: piecewise affine functions introduced in [1] to analyse WTGs,
and shrunk Difference Bound Matrices considered in [29] to analyse robustness
in timed automata. Last, we also study qualitative decision problems and close
an open problem on robust reachability, showing it is EXPTIME-complete for
general WTGs.
