Master Course on Optimal Control of Partial Differential Equations by Frédéric Bonnans

• Specialized course of the Optimization master , Inst. Pol. Paris and U. Paris-Saclay; AMS master, U. Paris-Saclay; Mathematics of modelization, Sorbonne Université, majeure COCV. Course codes: EN4726 (Optim.), EN5340 (AMS).

• Location: DMO (Lab. Math. Orsay).

• Schedule for 2023: February 14, 21, 28; March 07, 14, 21.

• Exam on March 28, 2-5pm (details given later).

As a motivation for the optimal control of PDEs, we will present the optimization problem of protein production based on gene expression thanks to the messenger RNA (mRNA). There is currently an increased interest for this type of technique, which however raises serious difficulties for the mathematical / numerical analysis. We will show how to express the original problem as an optimal control problem for the Fokker-Planck equation. The latter is a conservative, anisotropic parabolic PDE, with degeneracies at the boundary and control through the coefficients. We will discuss the well-posedness of this PDE (in appropriate weighted spaces) and the approximation of related optimal control problems by finite difference methods.

• D. Lunz, G. Batt, J. Ruess, J.F. Bonnans: Beyond the chemical master equation: Stochastic chemical kinetics coupled with auxiliary processes. PLoS Comput Biol 17(7), 2021.

USEFUL REFERENCES

• J.F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems. Springer, 2000.

• J.-L. Lions: Optimal Control of Systems Governed by Partial Differential Equations. Springer, 1971.

• J.-P. Raymond: Optimal Control of Partial Differential Equations. Lecture notes, U.P.S., Toulouse.

• F. Troeltzsch: Optimal Control of Partial Differential Equations - Theory, Methods and Applications. American Math. Society, 2010.