  There is a rising interest in Spatio-temporal systems described by Partial
Differential Equations (PDEs) among the control community. Not only are these
systems challenging to control, but the sizing and placement of their actuation
is an NP-hard problem on its own. Recent methods either discretize the space
before optimziation, or apply tools from linear systems theory under
restrictive linearity assumptions. In this work we consider control and
actuator placement as a coupled optimization problem, and derive an
optimization algorithm on Hilbert spaces for nonlinear PDEs with an additive
spatio-temporal description of white noise. We study first and second order
systems and in doing so, extend several results to the case of second order
PDEs. The described approach is based on variational optimization, and performs
joint RL-type optimization of the feedback control law and the actuator design
over episodes. We demonstrate the efficacy of the proposed approach with
several simulated experiments on a variety of SPDEs.
