  We study games with reachability objectives under energy constraints. We
first prove that under strict energy constraints (either only lower-bound
constraint or interval constraint), those games are LOGSPACE-equivalent to
energy games with the same energy constraints but without reachability
objective (i.e., for infinite runs). We then consider two kinds of relaxations
of the upper-bound constraints (while keeping the lower-bound constraint
strict): in the first one, called weak upper bound, the upper bound is
absorbing, in the sense that it allows receiving more energy when the upper
bound is already reached, but the extra energy will not be stored; in the
second one, we allow for temporary violations of the upper bound, imposing
limits on the number or on the amount of violations. We prove that when
considering weak upper bound, reachability objectives require memory, but can
still be solved in polynomial-time for one-player arenas; we prove that they
are in co-NP in the two-player setting. Allowing for bounded violations makes
the problem PSPACE-complete for one-player arenas and EXPTIME-complete for two
players.
