  Given a multi-modal dynamical system, optimal switching logic synthesis
involves generating the conditions for switching between the system modes such
that the resulting hybrid system satisfies a quantitative specification. We
formalize and solve the problem of optimal switching logic synthesis for
quantitative specifications over long run behavior. Each trajectory of the
system, and each state of the system, is associated with a cost. Our goal is to
synthesize a system that minimizes this cost from each initial state. Our paper
generalizes earlier work on synthesis for safety as safety specifications can
be encoded as quantitative specifications. We present an approach for
specifying quantitative measures using reward and penalty functions, and
illustrate its effectiveness using several examples. We present an automated
technique to synthesize switching logic for such quantitative measures. Our
algorithm is based on reducing the synthesis problem to an unconstrained
numerical optimization problem which can be solved by any off-the-shelf
numerical optimization engines. We demonstrate the effectiveness of this
approach with experimental results.
