This paper deals with formal controller synthesis for discrete-time dynamical systems. For a specification under the form of a discrete-time hybrid automaton, we aim at synthesizing controllers such that the trajectories of the closed-loop system are also trajectories of the hybrid automaton. We first show that the existence of an alternating simulation relation from the specification to the open-loop system is a necessary and sufficient condition for the existence of such controllers. Then, we propose an approach based on the use of symbolic (i.e. finite-state) abstractions of both the system and the specification. Effective computations are discussed for systems that are monotone and for specifications given by piecewise affine hybrid automata. We extend our approach to handle specifications with additional safety or reachability requirements. Finally, we illustrate our approach with examples from autonomous vehicle control
