The goal of this thesis is to provide an overview of the latest advances on reduced order methods for parametric optimal control governed by partial differential equations. Historically, parametric optimal control problems are a powerful and elegant mathematical framework to fill the gap between collected data and model equations to make numerical simulations more reliable and accurate for forecasting purposes. For this reason, parametric optimal control problems are widespread in many research and industrial fields. However, their computational complexity limits their actual applicability, most of all in a parametric nonlinear and time-dependent framework. Moreover, in the forecasting setting, many simulations are required to have a more comprehensive knowledge of very complex systems and this should happen in a small amount of time. In this context, reduced order methods might represent an asset to tackle this issue. Thus, we employed space-time reduced techniques to deal with a wide range of equations. We propose a space-time proper orthogonal decomposition for nonlinear (and linear) time-dependent (and steady) problems and a space-time Greedy with a new error estimation for parabolic governing equations. First of all, we validate the proposed techniques through many examples, from
the more academic ones to a test case of interest in coastal management exploiting the Shallow Waters Equations model.
Then, we will focus on the great potential of optimal control techniques in several advanced applications. As a first example, we will show some deterministic and stochastic environmental applications, adapting the reduced model to the latter case to reach even faster numerical simulations. Another application concerns the role of optimal control in steering bifurcating phenomena arising in nonlinear governing equations. Finally, we propose a neural network-based paradigm to deal with the optimality system for parametric prediction
