  Model-reduction techniques aim to reduce the computational complexity of
simulating dynamical systems by applying a (Petrov-)Galerkin projection process
that enforces the dynamics to evolve in a low-dimensional subspace of the
original state space. Frequently, the resulting reduced-order model (ROM)
violates intrinsic physical properties of the original full-order model (FOM)
(e.g., global conservation, Lagrangian structure, state-variable bounds)
because the projection process does not generally ensure preservation of these
properties. However, in many applications, ensuring the ROM preserves such
intrinsic properties can enable the ROM to retain physical meaning and lead to
improved accuracy and stability properties. In this work, we present a general
constrained-optimization formulation for projection-based model reduction that
can be used as a template to enforce the ROM to satisfy specific properties on
the kinematics and dynamics. We introduce constrained-optimization formulations
at both the time-continuous (i.e., ODE) level, which leads to a constrained
Galerkin projection, and at the time-discrete level, which leads to a
least-squares Petrov-Galerkin (LSPG) projection, in the context of linear
multistep schemes. We demonstrate the ability of the proposed formulations to
equip ROMs with desired properties such as global energy conservation and
bounds on the total variation.
