  We consider the optimal control problem of a general nonlinear
spatio-temporal system described by Partial Differential Equations (PDEs).
Theory and algorithms for control of spatio-temporal systems are of rising
interest among the automatic control community and exhibit numerous challenging
characteristic from a control standpoint. Recent methods focus on
finite-dimensional optimization techniques of a discretized finite dimensional
ODE approximation of the infinite dimensional PDE system. In this paper, we
derive a differential dynamic programming (DDP) framework for distributed and
boundary control of spatio-temporal systems in infinite dimensions that is
shown to generalize both the spatio-temporal LQR solution, and modern finite
dimensional DDP frameworks. We analyze the convergence behavior and provide a
proof of global convergence for the resulting system of continuous-time
forward-backward equations. We explore and develop numerical approaches to
handle sensitivities that arise during implementation, and apply the resulting
STDDP algorithm to a linear and nonlinear spatio-temporal PDE system. Our
framework is derived in infinite dimensional Hilbert spaces, and represents a
discretization-agnostic framework for control of nonlinear spatio-temporal PDE
systems.
