This paper studies the problem of control strategy synthesis for dynamical
systems with differential constraints to fulfill a given reachability goal
while satisfying a set of safety rules. Particular attention is devoted to
goals that become feasible only if a subset of the safety rules are violated.
The proposed algorithm computes a control law, that minimizes the level of
unsafety while the desired goal is guaranteed to be reached. This problem is
motivated by an autonomous car navigating an urban environment while following
rules of the road such as "always travel in right lane'' and "do not change
lanes frequently''. Ideas behind sampling based motion-planning algorithms,
such as Probabilistic Road Maps (PRMs) and Rapidly-exploring Random Trees
(RRTs), are employed to incrementally construct a finite concretization of the
dynamics as a durational Kripke structure. In conjunction with this, a weighted
finite automaton that captures the safety rules is used in order to find an
optimal trajectory that minimizes the violation of safety rules. We prove that
the proposed algorithm guarantees asymptotic optimality, i.e., almost-sure
convergence to optimal solutions. We present results of simulation experiments
and an implementation on an autonomous urban mobility-on-demand system.Comment: 8 pages, final version submitted to CDC '1
