  This study presents the extension of the data-driven optimal prediction
approach to the dynamical system with control. The optimal prediction is used
to analyze dynamical systems in which the states consist of resolved and
unresolved variables. The latter variables can not be measured explicitly. They
may have smaller amplitudes and affect the resolved variables that can be
measured. The optimal prediction approach recovers the averaged trajectories of
the resolved variables by computing conditional expectations, while the
distribution of the unresolved variables is assumed to be known. We consider
such dynamical systems and introduce their additional control functions. To
predict the targeted trajectories numerically, we develop a data-driven method
based on the dynamic mode decomposition. The proposed approach takes the
$\mathit{measured}$ trajectories of the resolved variables, constructs an
approximate linear operator from the Mori-Zwanzig decomposition, and
reconstructs the $\mathit{averaged}$ trajectories of the same variables. It is
demonstrated that the method is much faster than the Monte Carlo simulations
and it provides a reliable prediction. We experimentally confirm the efficacy
of the proposed method for two Hamiltonian dynamical systems.
