Kinetic and related models with applications in the natural sciences

Long time horizons and turnpike properties

Enrique Zuazua

DeustoTech & Universidad Autónoma de Madrid

PDE modelling employs time-depending systems to describe dynamics. On the other hand, with the aim of reducing the computational cost, frequently, time-dependence is avoided and steady state configurations are considered. In the PDE literature this is justified by proving, in a number of ways, convergence properties of time-evolving solutions towards the steady-state ones, as time tends to infinity. The validity of this model reduction process is then extrapolated to a number of contexts such as parameter-dependence problems, inverse and shape design, control, etc. Turnpike theory explores the validity of this reduction in the broad context of control and optimisation problems. In this lecture we shall briefly describe the state of the art and some challenging open problems arising in this field.