  This paper presents a compositional approach to specification-guided
abstraction refinement for control synthesis of a nonlinear system associated
with a method to over-approximate its reachable sets. Given an initial coarse
partition of the state space, the control specification is given as a sequence
of the cells of this partition to visit at each sampling time. The dynamics are
decomposed into subsystems where some states and inputs are not observed, some
states are observed but not controlled and where assume-guarantee obligations
are used on the uncontrolled states of each subsystem. A finite abstraction is
created for each subsystem through a refinement procedure starting from a
coarse partition of the state space, then proceeding backwards on the
specification sequence to iteratively split the elements of the partition whose
coarseness prevents the satisfaction of the specification. Each refined
abstraction is associated with a controller and it is proved that combining
these local controllers can enforce the specification on the original system.
The efficiency of the proposed approach compared to other abstraction methods
is illustrated in a numerical example.
