  In this work, we propose a novel approach for the continuous-time control
synthesis of nonlinear systems under nested signal temporal logic (STL)
specifications. While the majority of existing literature focuses on control
synthesis for STL specifications without nested temporal operators, addressing
nested temporal operators poses a notably more challenging scenario and
requires new theoretical advancements. Our approach hinges on the concepts of
signal temporal logic tree (sTLT) and control barrier function (CBF).
Specifically, we detail the construction of an sTLT from a given STL formula
and a continuous-time dynamical system, the sTLT semantics (i.e., satisfaction
condition), and the equivalence or under-approximation relation between sTLT
and STL. Leveraging the fact that the satisfaction condition of an sTLT is
essentially keeping the state within certain sets during certain time
intervals, it provides explicit guidelines for the CBF design. The resulting
controller is obtained through the utilization of an online CBF-based program
coupled with an event-triggered scheme for online updating the activation time
interval of each CBF, with which the correctness of the system behavior can be
established by construction. We demonstrate the efficacy of the proposed method
for single-integrator and unicycle models under nested STL formulas.
