# Seminario di Equazioni Differenziali e Applicazioni: “Simultaneous Controllability of Ensembles by Lie Algebraic Methods”

## Mercoledì 14 Marzo 2018, ore 14:30 - Sala Riunioni VII piano - Andrey Sarychev

ARGOMENTI: Seminars

Mercoledì 14 Marzo 2018 alle ore 14:30 in Sala Riunioni VII piano, Andrey Sarychev (Università di Firenze) terrà un seminario dal titolo “Simultaneous Controllability of Ensembles by Lie Algebraic Methods”.

Abstract
Over the last decade there have been a rise of interest regarding simultaneous controllability of ensembles (parameterized families) of nonlinear control systems
$$\frac{dx}{dt}=f^{\theta}(x,u),\quad\theta \in \Theta \subset \mathbb{R}^{\nu}$$
by a $\theta$-independent control $u(\cdot)$. Such problems arise for example, from a necessity to control a system with “structured uncertainty”, when some parameters of the system are subject to “dispersion”.
An alternative problem setting consists of simultaneous controlling an ensemble of points where the parameter affects the initial data
$$\frac{dx}{dt}=f(x,u),x(t_0)=\alpha(\theta),\quad\theta \in \Theta$$
The last case can be interpreted as controlled dynamics in the spaces of curves or surfaces.
Certainly one can consider mixed case when the parameter enters both the right-hand side and the initial data.
We aim at obtaining sufficient criteria of approximate controllability for these two models in the scope of Lie-algebraic approach of geometric control. We prove genericity of the property of “simultaneous controllability” for finite ensembles and provide infinite-dimensional extensions of Lie rank controllability criteria for the control of continual ensembles. We make a
comparison of the obtained results with existing versions of Rashevsky-Chow theorem for Banach spaces.

References
[1] A.Agrachev, Yu. Baryshnikov, A. Sarychev, Ensemble Controllability by Lie Algebraic Methods, ESAIM COCV, v. 22(2016), pp. 921-938.