We develop a discrete-time optimal control framework for systems evolving on Lie groups. Our work generalizes the original Differential Dynamic Programming method, by employing a coordinate-free, Lie-theoretic approach for its derivation. A key element lies, specifically, in the use of quadratic expansion schemes for cost functions and dynamics defined on manifolds. The obtained algorithm iteratively optimizes local approximations of the control problem, until reaching a (sub)optimal solution. On the theoretical side, we also study the conditions under which convergence is attained. Details about the behavior and implementation of our method are provided through a simulated example on T SO(3).
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.
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