AUTHOR=Schiassi Enrico , Calabrò Francesco , De Falco Davide Elia 

TITLE=Pontryagin Neural Networks for the Class of Optimal Control Problems With Integral Quadratic Cost

JOURNAL=Aerospace Research Communications

VOLUME=Volume 2 - 2024

YEAR=2024

URL=https://www.frontierspartnerships.org/journals/aerospace-research-communications/articles/10.3389/arc.2024.13151

DOI=10.3389/arc.2024.13151

ISSN=2813-6209

ABSTRACT=This work introduces Pontryagin Neural Networks (PoNNs) which are designed for learning the optimal control actions for the class of optimal control problems (OCPs) with integral quadratic cost. PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed and trained to tackle OCPs via applying the Pontryagin Minimum Principle (PMP). The PMP provides necessary optimality conditions, which result in a Two Points Boundary Value Problem (TPBVP) in the state-costate pair. This TPBVP is a system of Ordinary Differential Equations (ODEs). PoNNs learn the optimal control actions from the unknown solution of the arising TPBVP (i.e., states and costates), where the unknown solutions are modeled using Neural Networks. Moreover, the paper presents the estimate for the upper bounds on the generalization error of PoNNs in learning the OCP solutions for the class under consideration. The generalization error estimate is derived based of the choice and number of the training points, and the training error. The theoretical results are validated with numerical experiments on a benchmark linear and a benchmark nonlinear OCPs. Numerical studies show that PoNNs can be successfully applied to learn the control actions, in an open-loop fashion, for the class of OCPs considered. Furthermore, the results are compared against those generated with the commercial software GPOPS-II.PoNNs outperform the GPOPS-II both in terms of accuracy and computational time. In particular, the low computational time, makes PoNNs potentially attractive for real time applications.