Our team has made several contributions in the description of the limit behaviour, as the mesh sizes tend to zero, of **numerical schemes for wave and Schrödinger equations** from a control theoretical perspective. These results show that, in particular, filtering the high frequency numerical spurious solutions is necessary (and a good remedy) to assure the convergence of numerical schemes from a control perspective. These results also provide insight into the link between **conservative finite** and **infinite-dimensional dynamical systems** and their asymptotic behaviour.

#### Available tutorials:

#### The Optimal control on the Kuramoto adaptative coupling model with DyCon Toolbox

In this DyCon Toolbox tutorial, we present how to use OptimaControl enviroment to control a consensus that models the complex emergent dynamics over a given network.

#### Control for a semilinear heat equation and analogies with a collective behavior model

In this tutorial we will apply the DyCon toolbox to find a control to the semi-discrete semi-linear heat equation.

#### Optimal Control Problem with CasADi on null-controllability of the network system

The aim of this work is to use CasADi and IpOpt to simulate optimal control problem, which explains the structural controllability of the 2D heat equation. We use finite difference scheme with the uniform grid to test exact controllability of the 2D heat equation. After that, we delete several interactions between grid points and simulate the controllability with smaller number of controlled points.

#### Optimal strategies for guidance-by-repulsion model with IpOpt and AMPL

In this post, we use IpOpt and AMPL to simulate optimal controls on a nonlinear ODE system with unbounded interactions. The restriction and initial guess on the state variables are critical for this problem to operate minimization algorithm. From the data calculated from AMPL, we interpret and visualize it using Matlab.

#### The control on the Kuramoto model by handling one oscillator

We design the LQR controller and solve it with linearized model.

#### Stabilization of a collective behavior model

Summary of example objective The goal of this tutorial is to use LQR theory applied to a model of collective behavior. The model chosen shares a formal structure with the semidiscretization of the semilinear 1d heat equation.

#### Optimal control applied to collective behaviour

Guidance by repulsion model describing the behaviour of two agents, a driver and an evader

#### Kolmogorov equation

Various numerical approximation methods are discussed with the aim of recoving the large time asymptotic properties of the hypoelliptic Kolmogorov model