  This work recasts time-dependent optimal control problems governed by partial
differential equations in a Dynamic Mode Decomposition with control framework.
Indeed, since the numerical solution of such problems requires a lot of
computational effort, we rely on this specific data-driven technique, using
both solution and desired state measurements to extract the underlying system
dynamics. Thus, after the Dynamic Mode Decomposition operators construction, we
reconstruct and perform future predictions for all the variables of interest at
a lower computational cost with respect to the standard space-time discretized
models. We test the methodology in terms of relative reconstruction and
prediction errors on a boundary control for a Graetz flow and on a distributed
control with Stokes constraints.
