Introductory Lecture

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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
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