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We consider the optimal control problem of a general nonlinearspatio-temporal system described by Partial Differential Equations (PDEs).Theory and algorithms for control of spatio-temporal systems are of risinginterest among the automatic control community and exhibit numerous challengingcharacteristic from a control standpoint. Recent methods focus onfinite-dimensional optimization techniques of a discretized finite dimensionalODE approximation of the infinite dimensional PDE system. In this paper, wederive a differential dynamic programming (DDP) framework for distributed andboundary control of spatio-temporal systems in infinite dimensions that isshown to generalize both the spatio-temporal LQR solution, and modern finitedimensional DDP frameworks. We analyze the convergence behavior and provide aproof of global convergence for the resulting system of continuous-timeforward-backward equations. We explore and develop numerical approaches tohandle sensitivities that arise during implementation, and apply the resultingSTDDP algorithm to a linear and nonlinear spatio-temporal PDE system. Ourframework is derived in infinite dimensional Hilbert spaces, and represents adiscretization-agnostic framework for control of nonlinear spatio-temporal PDEsystems.

Spatio-Temporal Differential Dynamic Programming for Control of Fields