Control theory
Interpreting the dual Riccati equation through the LQ reproducing kernel
Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 199-204.

In this study, we provide an interpretation of the dual differential Riccati equation of Linear-Quadratic (LQ) optimal control problems. Adopting a novel viewpoint, we show that LQ optimal control can be seen as a regression problem over the space of controlled trajectories, and that the latter has a very natural structure as a reproducing kernel Hilbert space (RKHS). The dual Riccati equation then describes the evolution of the values of the LQ reproducing kernel when the initial time changes. This unveils new connections between control theory and kernel methods, a field widely used in machine learning.

Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.174
Pierre-Cyril Aubin-Frankowski 1

1. École des Ponts ParisTech and CAS, MINES ParisTech, PSL Research University, France
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Pierre-Cyril Aubin-Frankowski. Interpreting the dual Riccati equation through the LQ reproducing kernel. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 199-204. doi : 10.5802/crmath.174. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.174/

[1] Pierre-Cyril Aubin-Frankowski Linearly-constrained Linear Quadratic Regulator from the viewpoint of kernel methods (2020) (https://arxiv.org/abs/2011.02196)

[2] Viorel Barbu; Giuseppe Da Prato A representation formula for the solutions to operator Riccati equation, Differ. Integral Equ., Volume 5 (1992) no. 4, pp. 821-829 | MR 1167498 | Zbl 0757.49021

[3] Alain Bensoussan; Giuseppe Da Prato; Michel C. Delfour; Sanjoy K. Mitter Representation and Control of Infinite Dimensional Systems, Birkhäuser, 2007 | Article | Zbl 1117.93002

[4] David Luenberger Optimization by vector space methods, John Wiley & Sons, 1968

[5] Mario Micheli; Joan A. Glaunès Matrix-valued kernels for shape deformation analysis, Geometry, Imaging and Computing, Volume 1 (2014) no. 1, pp. 57-139 | Article | MR 3392140 | Zbl 1396.46025

[6] Saburou Saitoh; Yoshihiro Sawano Theory of Reproducing Kernels and Applications, Developments in Mathematics, 44, Springer, 2016 | MR 3560890 | Zbl 1358.46004

[7] Bernhard Schölkopf; J. Smola Alexander Learning with kernels : support vector machines, regularization, optimization, and beyond, MIT Press, 2002

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