### On the control of fluid flows by small forces

Alexander Shnirelman

Abstract.
Consider dynamical system described by the equation
$dx/dt=F(x)+f(t)$ where $f(t)$ is an external force (which can be
chosen arbitrarily). The system is controllable by small force, if for
any two states, $x_0$ and $x_1$, and any $\epsilon>0$ there exist
$T>0$ and a force $f(t)$ defined on $[0,T]$ such that the norm of
$f(t)$ (both pointwise and integrally) is less than $\epsilon$, and
the trajectory of the system with the external force $f(t)$, starting
at $t=0$ from $x_0$, finishes at $t=T$ at $x_1$. This property means
that there exist no reasonable integrals of the nonperturbed system.
In the case of infinite-dimensional systems, this property depends on
the norm in the phase space. Important example of a system
controllable by small force is the ideal incompressible fluid moving
on the 2-d torus and described by the Euler equations. This property
holds in the space $C$ of continuous velocity fields, but not in
$C^1$, and means the absence of "robust integrals" other than energy
and momentum. The proof is based on the generalized flows.