Ramanujan Auditorium

What is Control Theory in 2021? Can AI do Better?

Olivier Pironneau

Universite Pierre et Marie Curie, Laboratoire Jacques-Louis Lions

Until the twentieth century it was assumed that knowledge means control. Automatic control came in the sixties for electronics with Bellman's dynamic programming and Kalman's filter and received a boost in the eighties with robust and $H^\infty$ control. Will artificial intelligence algorithms change the practice of control drastically [4],[1].

Parallel Optimal Control [2], which dates from the calculus of variations of Hadamard and the Pontryajin principle [7], is a more functional approach to the optimization of systems. It is heavily used for the design of mechanical devices like airplanes (optimal shape design [6]) and the topological optimisation of materials [3].

Stochastic control[8] remained up to now a mathematical field except for the rare semi-analytical solutions as in the case of linear quadratic control. It is now computationally feasible and its applications to finance for instance. -- though challenged by deep neural networks -- are in daily use for risk assessment of bank's portfolios.

Finally, perhaps the most mathematically demanding is the mean-field type control [5] and its application to the Monge-Ampere problem.

As this is a colloquium talk, the problems and the main results will be stated only, without assuming any prior knowledge of theses sometimes difficult fields. Yet the talk is for a mathematically trained audience.

[1] D. Bertsekas. Reinforement Learning and Optimal Control. Athena Scientific, 2019.

[2] D.P. Bertsekas. Dynamic Programming and Stochastic Control. Athena Scientific, 2005.

[3] François Jouve Anca-Maria Toader Gregoire Allaire, Frederic De Gournay. Structural optimization
using topological and shape sensitivity via a level set method. Control and cybernetics, 2005.

[4] Simon Haykin. Kalman Filtering and Neural Network. 2001.

[5] J.M. Lasry and P.L. Lions. Mean field games” (pdf). japanese journal of mathematics. 2: 229–260. Japanese Journal of Mathematics., 2:229–260, 2007.

[6] O. Pironneau. Optimal Shape Design for Elliptic Systems. Springer, 1982.

[7] L. Pontriaguine, V. Boltianski, R. Gamkrelidze, and E. Michtchenko. Theorie Mathematique des
Processus Optimaux. Mir, 1974.

[8] Jiongmin Yong and Xun Yu Zhou. Stochastic Controls. Springer Verlag, NY, 1999.

Abstract pdf link: www.imsc.res.in/~anupdixit/IMSc_Pironneau_abstract.pdf

Poster for the talk: www.imsc.res.in/~anupdixit/IMSc_Colloquium.pdf