  The fundamental idea of this work is to synthesize reactive controllers such
that closed-loop execution trajectories of the system satisfy desired
specifications that ensure correct system behaviors, while optimizing a desired
performance criteria. In our approach, the correctness of a system's behavior
can be defined according to the system's relation to the environment, for
example, the output trajectories of the system terminate in a goal set without
entering an unsafe set. Using Metric Temporal Logic (MTL) specifications we can
further capture complex system behaviors and timing requirements, such as the
output trajectories must pass through a number of way-points within a certain
time frame before terminating in the goal set. Given a Mixed Logical Dynamical
(MLD) system and system specifications in terms of MTL formula or simpler
reach-avoid specifications, our goal is to find a closed-loop trajectory that
satisfies the specifications, in non-deterministic environments. Using an MILP
framework we search over the space of input signals to obtain such valid
trajectories of the system, by adding constraints to satisfy the MTL formula
only when necessary, to avoid the exponential complexity of solving MILP
problems. We also present experimental results for planning a path for a mobile
robot through a dynamically changing environment with a desired task
specification.
