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.
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