  The energetic optimization problem, e.g., searching for the optimal switch-
ing protocol of certain system parameters to minimize the input work, has been
extensively studied by stochastic thermodynamics. In current work, we study
this problem numerically with iterative dynamic programming. The model systems
under investigation are toy actuators consisting of spring-linked beads with
loading force imposed on both ending beads. For the simplest case, i.e., a
one-spring actuator driven by tuning the stiffness of the spring, we compare
the optimal control protocol of the stiffness for both the overdamped and the
underdamped situations, and discuss how inertial effects alter the
irreversibility of the driven process and thus modify the optimal protocol.
Then, we study the systems with multiple degrees of freedom by constructing
oligomer actuators, in which the harmonic interaction between the two ending
beads is tuned externally. With the same rated output work, actuators of
different constructions demand different minimal input work, reflecting the
influence of the internal degrees of freedom on the performance of the
actuators.
