Distributed Systems with Persistent Memory: Control and Moment Problems

We study controllability under the action of boundary deformation of systems with persistent memory which are important in engineering applications for example to viscoelasticity, nonfickian diffusion and thermal processes with memory. This problem has has been studied with several different methods and here we give an overview of some of the ideas used in this kind of study for systems of ``hyperbolic'' type, i.e. finite velocity of signal propagation (the ``parabolic'' case is far less studied, see papers by Barbu and Iannelli, Guerrero and Imanuvilo and Halanay and the present author). The methods that we are going to present are: operator methods (introduced by Belleni-Morante and used by the present author for control pourposes) are in Chap 2; a moment method approach to controllability, developed by the author, is in Chap. 5 (as an application, a source identification problem is studied in the final section). A circle of ideas introduced by Kim and which relays on the observation inequality is in Chap 6. The content of chapters 1, 3, 4 is as follows: Chapt. 1 treats a very simple example, in order to familiarize the readers with the subject of the book. Chap. 3 studies Riesz sequences and the moment problem, stressing the properties which are needed in the study of control problems for``hyperbolic'' type systems. Chap.4 recalls known properties of the controllability of the (memoryless) wave equation; The solutions of the problems proposed in every chapter can be downloaded from the author WEB page at the address http://calvino.polito.it/$\sim$lucipan/ricerca.html