In applied sciences one often observes the evolution of phenomena. They can differ in nature remarkably and their description may require finitely many as well as infinitely many variables. Moreover, the corresponding mathematical models can be formulated in terms of evolution equations, a structure where the dependence on time plays an essential role.
Fundamental questions concerning problems related to evolution equations include their well-posedness, when initial and (suitable) boundary conditions are prescribed (so that we have a so-called direct problem), as well as about the regularity in space and time of their solutions.
However, in several applications initial conditions are missing or only approximatively known, while measurements of the solution at different stages of the evolution are available. Moreover, often in applied models specific parameters entering the equation (or the boundary conditions) are unknown (or scarcely known). Consequently, many techniques have been developed to face these cases that are named generically inverse problems, or more exactly identification problems.
A third way to approach the subject is to influence the evolution of a given system by an external action called a control.
Control problems may be of different nature: one can try to lead a given system to a desired configuration in a finite or infinite time (position control) or try to optimize a performance criterion (optimal control).
For a long time, direct, inverse and control problem for evolution equations have been deemed practically independent of each other. However, in the past three decades it has become apparent that their interaction is fruitful mutually, so that they can be considered different aspects of the same topic. For instance, a priori Carleman estimates for solutions to PDEs – originally developed to study uniqueness – turned out later on to be fundamental tools to face exact controllability and identification problem as well as ill-posed problems.
The main aim we have pursued in these past five years has been to collect experts in the quoted research fields and so to stimulate the development of new ideas and contributions in these areas.
For this purpose, two international conferences were co-organized with Prof. Piermarco Cannarsa: the first one Workshop on Direct, Inverse and Control Problems for PDEs (DICOP07) was held in Rome in June 2007 at the Istituto Nazionale di Alta Matematica (INdAM), while the latter Meeting on Direct, Inverse and Control Problems for PDEs (DICOP08) was held in September 2008 at the Palazzone in Cortona.
The surprising success of the first event pleasantly forced us to organize the latter, stimulating the lecturers to improve their original contributions delivered during the meeting to a paper form for the benefit of the scientific community. The contributions so obtained were subjected to peer-review and then published in 2011 in the volume New trends in Direct, Inverse and Control Problems for Evolution Equations, edited by P. Cannarsa, C. Cavaterra, A. Favini, A. Lorenzi and E. Rocca in Direct and Continuous Dynamical Systems, Series S.
The success of these events stimulated us to organize the meeting PDEs, Inverse and Control Problems held in September 2010 in Bologna at the Department of Mathematics, financed by the Italian Ministry of Education and University and the Universities of Bologna and Milan.
The present volume collects 10 contributions of the speakers of the Bologna meeting that we have chosen for their particular interest and novelty. All the papers have been peer-reviewed according to the standard procedure of Applicable Analysis.
Finally, we want to thank all the authors for the effort required to write their excellent papers as well as the editors of the journal for their interest in our volume. Particular thanks to Roberto Triggiani for his continuous help and assistance.
Guest Editors
Angelo Favini
Università degli Studi di Bologna
Alfredo Lorenzi
Università degli Studi di Milano