**Carlos Esteve Yagüe** holds a Postdoctoral position at the ERC Advanced Grant project DyCon under the supervision of Prof. Enrique Zuazua (UAM and DeustoTech). He earned his PhD in analysis of nonlinear parabolic partial differential equations under the supervision of Prof. Philippe Souplet (Université Paris 13). His main research interests are singularity formation for parabolic PDE, Hamilton-Jacobi equations, optimal control theory and differential games.

#### Author's contribution:

#### Multilevel Selective Harmonic Modulation via Optimal Control

The Multilevel Selective Harmonic Modulation problem is recast under the perspective of optimal control

#### Q-learning for finite-dimensional problems

In this tutorial we show how to implement the Q-learning algorithm in simple settings where the state-space and the control-space are finite. In this case, the Q-function can be represented by a table, and therefore, it belongs to a finite-dimensional vector space. We illustrate the algorithm by solving the problem of finding the shortest path between any arbitrary point in a discretized domain and a target area. We then consider the same problem in a domain affected by a potential.

#### Inverse problem for Hamilton-Jacobi equations

In this tutorial we study the inverse design problem for time-evolution Hamilton-Jacobi equations. More precisely, for a given observation of the viscosity solution at time $T>0$, we construct all the possible initial data that could have led the solution to the observed state. We note that these initial data are not in general unique.

#### Touchdown localization for the MEMS problem with variable dielectric permittivity

In this tutorial we study the localization of touchdown points in a mathematical model for micro-electro-mechanical systems (MEMS) with variable dielectric permittivity. We consider a device consisting of two conducting plates, connected to an electric circuit. The upper plate is rigid and fixed while the lower one is elastic and fixed only at the boundary. When a voltage (difference of potential between the two plates) is applied, the lower plate starts to bend and, if the voltage is large enough, the lower plate eventually touches the upper one. This is called touchdown phenomenon and our aim is to control the localization of touchdown points in the device by a suitable choice of the dielectric permittivity of the material.