  In this contribution we propose reduced order methods to fast and reliably
solve parametrized optimal control problems governed by time dependent
nonlinear partial differential equations. Our goal is to provide a tool to deal
with the time evolution of several nonlinear optimality systems in many-query
context, where a system must be analysed for various physical and geometrical
features. Optimal control can be used in order to fill the gap between
collected data and mathematical model and it is usually related to very time
consuming activities: inverse problems, statistics, etc. Standard
discretization techniques may lead to unbearable simulations for real
applications. We aim at showing how reduced order modelling can solve this
issue. We rely on a space-time POD-Galerkin reduction in order to solve the
optimal control problem in a low dimensional reduced space in a fast way for
several parametric instances. The proposed algorithm is validated with a
numerical test based on environmental sciences: a reduced optimal control
problem governed by viscous Shallow Waters Equations parametrized not only in
the physics features, but also in the geometrical ones. We will show how the
reduced model can be useful in order to recover desired velocity and height
profiles more rapidly with respect to the standard simulation, not losing
accuracy.
