  Correlated with the trend of increasing degrees of freedom in robotic systems
is a similar trend of rising interest in Spatio-Temporal systems described by
Partial Differential Equations (PDEs) among the robotics and control
communities. These systems often exhibit dramatic under-actuation, high
dimensionality, bifurcations, and multimodal instabilities. Their control
represents many of the current-day challenges facing the robotics and
automation communities. Not only are these systems challenging to control, but
the design of their actuation is an NP-hard problem on its own. Recent methods
either discretize the space before optimization, or apply tools from linear
systems theory under restrictive linearity assumptions in order to arrive at a
control solution. This manuscript provides a novel sampling-based stochastic
optimization framework based entirely in Hilbert spaces suitable for the
general class of \textit{semi-linear} SPDEs which describes many systems in
robotics and applied physics. This framework is utilized for simultaneous
policy optimization and actuator co-design optimization. The resulting
algorithm is based on variational optimization, and performs joint episodic
optimization of the feedback control law and the actuation design over
episodes. We study first and second order systems, and in doing so, extend
several results to the case of second order SPDEs. Finally, we demonstrate the
efficacy of the proposed approach with several simulated experiments on a
variety of SPDEs in robotics and applied physics including an infinite
degree-of-freedom soft robotic manipulator.
