Spatio-Temporal Differential Dynamic Programming for Control of Fields

TL;DR

This paper introduces a novel infinite-dimensional differential dynamic programming framework for controlling nonlinear spatio-temporal PDE systems, offering a discretization-agnostic approach that generalizes existing methods and ensures global convergence.

Contribution

It develops a new DDP-based control framework for infinite-dimensional PDE systems, extending finite-dimensional DDP and spatio-temporal LQR solutions, with proven convergence and practical numerical methods.

Findings

Framework applies to linear and nonlinear PDE systems

Proven global convergence of the control algorithm

Demonstrated effectiveness through numerical experiments

Abstract

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.