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

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.