PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 1, Pages 117–127 S 0002-9939(05)08340-1 Article electronically published on August 22, 2005 A CONSTRUCTION OF A ROBUST FAMILY OF EXPONENTIAL ATTRACTORS STEFANIA GATTI, MAURIZIO GRASSELLI, ALAIN MIRANVILLE, AND VITTORINO PATA (Communicated by David S. Tartakoff) Abstract. Given a dissipative strongly continuous semigroup depending on some parameters, we construct a family of exponential attractors which is robust, in the sense of the symmetric Hausdorff distance, with respect to (even singular) perturbations. 1. Introduction Exponential attractors for strongly continuous semigroups were first introduced in [2]. The motivation comes from the lack of effective information given by global attractors, which present two major drawbacks, since they do not provide an actual control of the convergence rate of trajectories, and, consequently, they might be quite unstable with respect to perturbations (cf. [11]). The original technique to build exponential attractors was developed in Hilbert spaces, and it heavily relied on the use of orthogonal projections. Later, in [1, 3] new methods were introduced, that work in a Banach space setting. A further interesting issue is to consider not just a single semigroup, but rather a family of semigroups depending on certain parameters. It is then of some impor- tance to have results establishing good stability properties of the related exponential attractors, in dependence of the parameters. Significant achievements in this direc- tion have been obtained in [4, 10] and, especially in [6], where the case of a singular perturbation has been successfully treated. In this paper, we provide a stability result of the same kind, which takes into account the case of singular perturbations. What mostly motivated our investi- gation is that, in order to apply the methods of [6] to concrete problems arising from PDE, one has to construct some Banach spaces that are not natural, in the sense that they are not the spaces suggested by the equations. Besides, it might be quite difficult to understand which is the correct construction that works in a given setting. On the contrary, our technique provides a sort of “machinery” that can be automatically applied, using the correct spaces where the solutions live. As a Received by the editors December 14, 2003. 2000 Mathematics Subject Classification. Primary 37L25, 37L30. Key words and phrases. Strongly continuous semigroups, robust exponential attractors, fractal dimension. This research was partially supported by the Italian MIUR FIRB Research Project Analisi di Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metodologici, Modellistica, Applicazioni. c 2005 American Mathematical Society 117 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use