  The aim of this work is to present a model reduction technique in the
framework of optimal control problems for partial differential equations. We
combine two approaches used for reducing the computational cost of the
mathematical numerical models: domain-decomposition (DD) methods and
reduced-order modelling (ROM). In particular, we consider an optimisation-based
domain-decomposition algorithm for the parameter-dependent stationary
incompressible Navier-Stokes equations. Firstly, the problem is described on
the subdomains coupled at the interface and solved through an optimal control
problem, which leads to the complete separation of the subdomain problems in
the DD method. On top of that, a reduced model for the obtained optimal-control
problem is built; the procedure is based on the Proper Orthogonal Decomposition
technique and a further Galerkin projection. The presented methodology is
tested on two fluid dynamics benchmarks: the stationary backward-facing step
and lid-driven cavity flow. The numerical tests show a significant reduction of
the computational costs in terms of both the problem dimensions and the number
of optimisation iterations in the domain-decomposition algorithm.
