  Sums-of-squares (SOS) optimization is a promising tool to synthesize
certifiable controllers for nonlinear dynamical systems. Building upon prior
works, we demonstrate that SOS can synthesize dynamic controllers with bounded
suboptimal performance for various underactuated robotic systems by finding
good approximations of the value function. We summarize a unified SOS framework
to synthesize both under- and over- approximations of the value function for
continuous-time, control-affine systems, use these approximations to generate
approximate optimal controllers, and perform regional analysis on the
closed-loop system driven by these controllers. We then extend the formulation
to handle hybrid systems with contacts. We demonstrate that our method can
generate tight under- and over- approximations of the value function with
low-degree polynomials, which are used to provide stabilizing controllers for
continuous-time systems including the inverted pendulum, the cart-pole, and the
quadrotor as well as a hybrid system, the planar pusher. To the best of our
knowledge, this is the first time that a SOS-based time-invariant controller
can swing up and stabilize a cart-pole, and push the planar slider to the
desired pose.
