Minisimposium "Control Problems for Fluidodynamic Models"

The minisimposium was part of the "6th International Congress on Industrial and Applied Mathematics (ICIAM 2007)'', held in ETH e University of Zurich, on July 2007. It was aimed to put together researchers working in the field of control problem for nonlinear hyperbolic PDEs and in modelization of traffic flow problems and supply chains. Abstract: The theory of weak solutions of hyperbolic conservation laws received many foundamental contributions in the last years. In particular, the well-posedness theory for the Cauchy problem and for the mixed initial-boundary value problem is now well established in the case of systems in one space dimension. On the other hand, the study of these equations from the point of view of control theory is still at an early stage. The interest on such problems is motivated by applications to traffic flow models, multicomponent chromatography, as well as in problems of oil resevoir simulation and gas dynamic. Another related interesting area of research regards the control of equations over networks, to address application domain as car traffic flow, telecommunication, irrigation channels, etc. When studying the effect of the boundary data treated as controls acting on the solutions of a conservation law one cannot expect to achieve in general complete controllability results within the space of discontinuous weak solutions, due to the particular wave-front structure of the solutions of such systems. It is then more appropriate to consider the problem of asymptotic stabilization of an hyperbolic system with boundary controls. Of particular interest for applications are general boundary controllability and stabilizability problems where the control acts only on some of the boundary conditions. Another relevant direction in which it is being pursued the investigation of hyperbolic control problems is the study of necessary conditions for the optimality of a weak solution where the controls may act through the boundary conditions as well as through the source term of the balance laws. Here, the main source of difficulties stems from the fact that the input-to-trajectory map that associates to a given control the corresponding solution may not be differentiable in any natural Banach space. For this reason to tackle such problems it has been crucial to introduce a suitable variational structure on the flow generated by conservation laws.