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%\English

\begin{document}
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\large

\begin{center} {\bf
THE BROCKETT STABILIZATION PROBLEM
}
\end{center}

\centerline{\it Gennadij A. Leonov}

\medskip

\centerline{Dept. of Mathematics and Mechanics}
\centerline{St.Petersburg State University}
\centerline{2, Bibliotechnaya pl., Peterhof}
\centerline{St.Petersburg 198504, Russia}
\centerline{leonov@math.spbu.ru}

\bigskip

\bigskip

{
\centerline{Abstract}

\smallskip

R. Brockett has formulated in book ``Open problems in
mathematical systems and control theory" (Springer 1999)
the following problem.

Given  a triple constant matrices $A,B,C$ under what circumstances does
there exist a time-dependent matrix $K(t)$ such that the  system
$$
\f{dx}{dt}=Ax+BK(t)Cx,\quad x\in\br^n
\eqno(1)
$$
is asymptotically stable.

In a number of cases the answer to this question can be found by
means of the following results.

\medskip

{\bf Theorem 1.} {\it
Let there exist matrices $K_1$ and $K_2$ satisfying the following
conditions:

$1)$ a matrix $BK_1C$ has $n-1$ eigenvalues with negative real
parts and $\det BK_1C=0$,

$2)$ for a vector $u\neq0$, satisfying an equality $BK_1Cu=0$, and
for a certain number $\la$, the vector-function
$$
\exp\big[(A+BK_2C+\la I)t\big]u
$$
is periodic.

Then there exists a periodical matrix $K(t)$ such that system $(1)$
is asymptotically stable.
}

\medskip

{\bf Theorem 2.} {\it
Let $n=2$ and there exist matrices $K_1$, $K_2$ such that
the following conditions hold

$1)$ $\det BK_1C=0$, $\ \hbox{\rm Tr}\,BK_1C\neq0$,

$2)$ a matrix $A+BK_2C$ has complex eigenvalues.

Then there exists a periodic matrix $K(t)$ such that system $(1)$
is asymptotically stable.
}

\medskip

{\bf Corollary.}
Let $n=2$, $B$ is a column vector, $C$ is a row vector,
$K(t)$ is a function, the transfer function has the form
$$
W(p)=C(A-pI)^{-1}B=\f{p+\ga}{p^2+\al p+\beta}\,,
$$
where $\ga\leq0$, $\al\ga>\beta$.

If
$$
\ga^2-\al\ga+\beta>0
$$
then there exists a periodic function $K(t)$ such that system (1)
is asymptotically stable.

If
$$
\ga^2-\al\ga+\beta<0
$$
then there do not exist functions $K(t)$, for which system (1) is
asymptotically stable.

{\bf Theorem 3.} {\it
Let $n=3$, $B$ is a column vector, $C$ is a row vector, $K(t)$ is
function,  $CB\neq0$ and there exist a numbers $K_1,K_2$ satisfying the
following conditions:

$1)$ a matrix $A+K_1BC$ has two complex eigenvalues and one
negative eigenvalue,

$2)$ function
$$
C\exp[(A+K_2BC)t]B
$$
has at least one zero on interval $(-\infty,0)$.

Then there exists a periodic function $K(t)$ such that system
$(1)$ is asymptotically stable.
}
}

\end{document}

