Introductory Lecture

MEETING ALAIN
VILMOS KOMORNIK
It is a great privilege for me to say a few words about Alain at the
beginning of this special meeting. I would have never imagined such
a possibility when I first met him in November 1982 at the lectures of
Jacques-Louis Lions at the Collège de France, just some days after my
first arrival to France. We were both interested in understanding the
recent solution by Bensoussan of an optimal control problem of Lions.
His openness and simplicity helped me a lot in filling many gaps in my
knowledge of partial differential equations during my stay.
One year later I met him again at a conference in Prague. Our
research collaboration started there and was performed by exchanging
many letters in that pre-internet era. His intriguing questions about
the pointwise oscillation of the solutions of the wave equation led us
to surprising paradoxical results. His approach seemed to me more
Hungarian than French. He was very curious, he raised simple and
natural questions which resisted the classical general methods. New
ideas had to be found, and then they led to numerous applications in
solving other, sometimes seemingly unrelated problems. This way of
working was characteristic of Fejér, Pólya, Turán and Erdős.
Our correspondance continued for years. Alain probably does not
know that his letters meant for me almost the only connection with
active mathematics during a whole year that I spent in a remote part
of Venezuela in 1985–86. And that his arrangement to invite me to
Paris for three months at the end of 1986 was crucial for me to resume
my research. Incidentally, I happened to return to Paris just during
the now historic lectures of Jacques-Louis Lions on exact controllability,
and these lectures influenced my subsequent research for many years.
Let me mention some more examples where his ideas greatly stimulated
my own work. In two preprints in 1978, he introduced strangely
looking new integral inequalities, similar to Gronwall’s lemma but with
a reverse sign. It turned out ten years later that they provide sharp decay
rate estimates for a large class of dissipative evolutionary problems;
they are simpler and more efficient than the usual Liapunov function
method. By the way, many papers on boundary stabilization were influenced
by his joint work with Zuazua on the Liapunov method for
internal stabilization. These results still stimulate much research: most
recently Alabau–Boussouira proved a very useful extension of his integral
inequalities.
Date: March 23rd, 2009.
1

2 VILMOS KOMORNIK
In another work he needed a constructive proof of an Ingham–Parseval
type result of Ball and Slemrod. He accomplished this by a clever
and original argument. Hhis elementary lemma proved to be of great
importance in the determination of the critical time of exact controllability:
in a large number of situations it allowed one to eliminate
elaborate and very technical arguments based on microlocal analysis,
Carleman estimates or Holmgren type uniqueness theorems. In collaboration
with Baiocchi and Loreti we have used many times Haraux’s
method for various systems.
Completing the important results of Lasiecka–Triggiani, Ho and Lions
on the exact controllability of the wave equation, Bardos, Lebeau
and Rauch essentially characterized the control domains by their socalled
geometric control condition related to reflected rays. Alain made
two important contributions to this domain. First, he proved that by
applying very singular controls this condition is not any more necessary.
Secondly, he proved by ingeniously simple Ingham type arguments that
the geometric control condition is not necessary for rectangular plates,
refuting a conjecture that the control domains are the same for membranes
and plates. His result was improved by Jaffard, and these results
stimulated much research in clarifying the role of singularities for this
type of phenomenon.
Looking at the list of publications of Alain Haraux, we realize in how
many different fields he has worked. I can only mention a few results
here.
• He started his research in collaboration with Brezis on maximal
monotone operators.
• He obtained, partly in collaboration with Biroli and Otani,
many interesting results on periodic or quasi-periodic solutions
to parabolic or hyperbolic problems.
• He established global existence and uniqueness results for nonlinear
Schrödinger equations in collaboration with Cazenave,
and non-uniqueness theorems for nonlinear parabolic equations
in collaboration with Weissler.
• Simultaneously with Hale and Ladyzenskaya, he discovered an
important property of attractors of dynamical systems.
• He studied the global behavior of solutions to various complex
nonlinear hyperbolic, parabolic and finite-dimensional systems
without monotonicity but having some dissipativity.
• In collaboration with Jendoubi they succeeded in adapting a
powerful method of Lojasiewicz and Simon to prove the asymptotic
stability for some dissipative hyperbolic systems. In
one of their papers they discovered a hidden connection among
the approaches of Lojasiewicz–Simon, Aulbach, Zelenyak and
Hale–Raugel. With Jendoubi and Kavian he also estimated the
decay rate of semilinear parabolic systems by this method.

MEETING ALAIN 3
I am apologizing for not having enough time to name all of his more
than thirty collaborators, but I am sure that most of the important
works will be covered during this conference. This will not be easy because
his publications contain a multitude of original observations and
ideas. His innocently-looking questions often led to elaborate theories
and to many other questions, so that instead of clearing out a particular
research field, his works opened up new directions with promising
problems. This is why he had and has so many students and collaborators.
In the last ten years our encounters become more sporadic: e-mail
and telephone became cheaper than the train tickets and the hotels in
Paris. We still had some nice meetings. Once, returning from Oberwolfach,
he had to change the train in Strasbourg. He hesitated to call
me to meet because he only had one hour to stay. Luckily, I arrived
to the railway station just at that moment with Claudio Baiocchi who
was about to return to Rome from Strasbourg. We had a very nice
conversation all together.
I would like end with two further comments on Alain’s working style.
As I told, his openness and simplicity make it easy for young researchers
to discuss with him. But at the same time, he maintains a
high scientific rigor. His students have to work hard in order to get
recognition. But those who succeed become independent, good and
passionate researchers.
By reading his papers, his books or by listening to his informal explanations
or official talks, I was often struck by the clarity and simplicity
of his expositions. I hope that this conference will reinforce his scientific
influence and that numerous present and future colleagues will
profit of his help, insight and teaching for many years to come.
University of Strasbourg