So welcome everyone to the seminar.

We have also Alexander Trouyev from the Max Plan Institute for Dynamic Complexity Inter-System

that's in Magdeburg.

And today he will be speaking about stabilization problems with oscillating input on the non-linear

comparability conditions.

Please Alexander, we're looking forward.

Okay, thank you very much for the introduction.

It's my great pleasure to be here at the seminar of Applied Analysis Group and at least virtually.

And today I would like to give a talk about stabilization with oscillating inputs.

And before presenting the main results, I would like to present a brief scheme about

this topic.

So I would like to start with some motivation.

Why do we need this class of systems?

And we try to establish the relation between controllability and stabilizability.

To do so, we will apply fast oscillating controls, time-variant feedback controls.

And there will be some examples coming from non-holonomic mechanics, from mechanics of

rigid bodies.

And I will conclude with some examples from hydrodynamics, namely with the Euler equations

and Navier-Stokes equations, at least with their finite dimensional approximations.

So what we will be dealing with is considered the following derivation.

Here on this slide, you see three classes of essentially non-linear systems.

What do I mean by essentially non-linear?

I mean that if a linear rise, which is controllable or stabilizable.

First of all, you see the well-known unicycle example where we have rolling without sleeping

condition and X1, X2, X3 are respectively the coordinates of the center of contact of

the wheel with the plane and the angle between the wheel and certain fixed direction.

U1 is the velocity control and U2 is the angular velocity control.

So it's clear that the system has zero equilibrium with zero controls.

If linear rise dynamics, we see that the linear rise dynamics is neither controllable nor stabilizable.

The same observation also happens with another example where I presented Euler equations

in rigid body dynamics.

So it may be considered as a model of a rotating satellite actuated by two gas jets.

You see the terms in red are quadratic here.

So if you linearize the system at the origin, you don't have a system which you can stabilize

by the usual linear technique.

And in fact, it is not controllable by the linear approximation.

Also similar observation applies to a class of essentially nonlinear models in hydrodynamics.

On the bottom part of this slide, you see the Navier-Stokes equations or Euler equation

in particular, if you put the viscosity parameter being zero.

Also there are quadratic terms marked in red here.

If we neglect those terms, we cannot control the system and we cannot stabilize it in a

classical sense.

So those three examples were just for motivation that we need to use some kind of nonlinear

tools for the stabilization at least of these class of mechanical systems.

Formally, I will address a general stabilization scheme, general stabilization problem in this

talk.

Suppose that we have a nonlinear controller fine system of form sigma.

In general, it may contain drifter F0.

And in general, we assume that there is a equilibrium X equals zero when controls are

all zero.