**Numerical methods, modeling and control of non-linear dynamics**

I am interested in the study of self-organized system, from a numerical and analytical point of view.

Usually these models take in account the evolution of a large number of interacting individuals interacting each other, and they can be described by an ODEs system with the following structure $$\left\{\begin{array}{l}\dot{x}_i=v_i\\\dot{v}_i=\sum_{j=1}^Nm_jF(x_i,x_j,v_i,v_j)\end{array}\right.\qquad i=1,\ldots, N$$ where $(x_i,v_i)\in\mathbb{R}^{2d}$, $d\geq 1$ and $F(\cdot)$ describes the possibly non-linear interaction among agents.

Solving such systems implies a huge computational effort when the number of agents $N$ is large, and this cost increase further for large values of the dimensional parameter $d$.

A first approach to reduce such computational effort, consists in the derivation of a

*mesoscopic description*of the system. Thus the density function $f=f(x,v,t)$ of agents at time $t$ in position $(x,v)\in\mathbb{R}^{2d}$ is introduced, whose evolution is described by the following kinetic equation $$\partial_t f +v\cdot\nabla_x f=-\nabla_v\cdot\left(\mathcal{F}[f]f\right).$$ One of the main goal of my research consists in the development of efficient numerical methods for the solution of such models. A further direction we are investigating is the action of few individuals among a large set of individuals, examples run from the study of predator-prey models in biological system, to the influence of opinion leaders in a social network.

For more insights look at the following links:

- swarm-interacting-with-few-individuals/
- invisible-control-of-self-organizing-leaving-an-unknown-enviroment/
- Stability analysis of swarming dynamics/