%%%%%%%%title of contribution
\Atitle{Dissipation Inequalities in Systems Theory: An Introduction and Recent Results}

%%%%%%%%name of authors connected by "and"
\Aauthor{\underline{Frank Allg{\"o}wer}, Tobias Raff and Christian Ebenbauer}

%%%%%%%%abbreviation of authors names for running header
\renewcommand{\rauthor}{F. Allg{\"o}wer}

%%%%%%%affiliations and e-mail
\Aaddress{%
Institute for Systems Theory and Automatic Control\\
University of Stuttgart\\
Pfaffenwaldring 9\\
70550 Stuttgart
Germany\\
{\tt allgower@ist.uni-stuttgart.de}}

%%%%%%%please enter some keywords describing your contribution
%\Akeywords{some keywords}

%%%%%%%abstract of contribution
\Aabstract{%
% Some abstract, see~\cite{lit1} for more details
Lyapunov function techniques have
received constantly high interest in applied mathematics and in particular in
systems and control theory \cite{Lyp-92,Yos-66,Hah-67} over the last hundred years.
The main reasons for this interest are simplicity, intuitive appeal, and universality of these techniques.
%
Today, there is no doubt that Lyapunov functions techniques are the main tools to be used
when one is faced with a stability or stabilization problem.
%
In the analysis and design of control systems, however, there are usually other important
requirements beside{s} stability which have to be taken into account.
%For example robustness or performance requirements.
%
{Therefore, it is natural to ask the following question: Is it possible to generalize the
ideas of Lyapunov function techniques in order
to address for example robustness and performance issues in control systems?}
%
Such a generalization is indeed possible and has lead to the powerful concept of dissipativity
and dissipation inequalities.
%
 Dissipativity has been introduced by Willems \cite{Wil-72}
and is motivated by the concept of passivity, a paradigm from electrical network theory
which relates the stored energy in an electrical network with the supplied {energy} into the network.
%
Alternatively, one can say that
the basic idea behind dissipativity is to generalize the concept of Lyapunov functions techniques
to systems with inputs and outputs.
%
Over the past decades, dissipativity turned out to be an extremely
useful concept in systems and control theory with plenty of applications in theory and practice.
%

Like in Lyapunov theory the biggest problem in applications is the construction
of a storage function, which is the generalization of the Lyapunov function.
However for the important class of polynominal systems, i.e. systems with
polynomial nonlinearities, recent advances in the area of computational
semialgebraic geometry, namely semidefinite programming and the sum
of squares decomposition, allow a reliable and efficient solution in many
cases~\cite{Par-00,Ebe-All-06c}.

In this talk we will give a brief historical perspective and an introduction to the
system theoretic concept of dissipation inequalities. We will present exemplary
recent results on the stability analysis of nonlinear differential algebraic equation
systems, minimum phase analysis, and nonlinear feedback and observer
design that are based on novel dissipation inequalities and will discuss
questions concerning the computation of the storage functions. The methods
will be demonstrated and critically assessed with various examples from
engineering and systems biology.



%A list of references might be at the end of the abstract.  When referring to
%them in the text, type the corresponding reference number
%between brackets \cite{lit1}.  References should be given in the standard
%style as follows.

\begin{thebibliography}{XXX}
%\bibitem{lit1}
%Author, ``Title'', {\em Journal}, {\bf No.}, pp.XXX-XXX, (Year) .
%
\bibitem{Lyp-92}
A.~M. Lyapunov,
``The General Problem of the Stability of Motion'',
{\em CRC Press}, (1992) .


\bibitem{Yos-66}
T.~Yoshizawa,
``Stability theory by Liapunov's second method'', {\em Math. Soc. Japan Tokyo}, (1966).

\bibitem{Hah-67}
W.~Hahn,
``Stability of Motion'',
{\em Springer}, (1967).

\bibitem{Wil-72}
J.~C. Willems,
``Dissipative dynamical systems - {P}art {I}: General theory'', 
{\em Arch. Ratl. Mech. and Analysis.}, {\bf 45:321--351}, (1972).

\bibitem{Ebe-All-06c}
C.~Ebenbauer and F.~Allg{\"o}wer,
``A dissipation inequality for the minimum phase property'', 
{\em IEEE Transactions on Automatic Control}, {\bf 53}, February (2008).

\bibitem{Par-00}
P.A. Parrilo, 
``Structured Semidefinite Programs and Semialgebraic Geometry
  Methods in Robustness and Optimization'',
{\em PhD thesis, California Institute of Technology}, (2000).

\end{thebibliography}
}