  This work studies the linear approximation of high-dimensional dynamical
systems using low-rank dynamic mode decomposition (DMD). Searching this
approximation in a data-driven approach is formalised as attempting to solve a
low-rank constrained optimisation problem. This problem is non-convex and
state-of-the-art algorithms are all sub-optimal. This paper shows that there
exists a closed-form solution, which is computed in polynomial time, and
characterises the l2-norm of the optimal approximation error. The paper also
proposes low-complexity algorithms building reduced models from this optimal
solution, based on singular value decomposition or eigen value decomposition.
The algorithms are evaluated by numerical simulations using synthetic and
physical data benchmarks.
