Relation to DMDc?

[gramian]

How (if at all) is Section 2 in this code's associated preprint "Data-Driven Reachability Analysis Using Matrix Zonotopes"
related to dynamic mode decomposition with control (see references below)? The use of the X+, X- and U- looks somewhat similar.

Proctor, J.L., Brunton, S.L., Kutz, J.N.: Dynamic mode decomposition with control.
SIAM J. Appl. Dyn. Syst. 15(1), 142--161 (2016). https://doi.org/10.1137/15M1013857

Proctor, J.L., Brunton, S.L., Kutz, J.N.: Generalizing Koopman Theory to Allow for Inputs and Control.
SIAM J. Appl. Dyn. Syst. 17(1), 909--930 (2018). https://doi.org/10.1137/16M1062296

[aalanwar]

Dear Christian,

thanks for your email and for your interest in our work.

Even though I am not an expert in dynamic mode decomposition myself, I will try to point out some relations and differences between our work and the references that you mentioned.
If you look into the first reference that you sent, then you see that equation (2.4) (and with input equation 3.14) are a simple result of having (input-) state data and fitting an A (and B) matrix via least-squares. The authors state it also themselves - and this relationship can be found in many more references (e.g. https://arxiv.org/pdf/1903.06842.pdf equations (8)-(10) ).
This first reference that you pointed out provide a specific viewpoint on this least-squares approach (and maybe improve it?) via SVD / DMD.
At first sight, a big difference between that reference and our work is that in the reference via least-squares or via SVD the goal is to find the 'best fitting' A (and B) that can explain the data - which probably approximates the true A and B closely but it will most probably not be the correct A and B given some noise in the data. We on the other hand are trying to estimate the whole set of A and B matrices that could possibly explain the data given some bound on the noise. We are then sure that the correct A and B lies within this set, which allows us to give guarantees on the reachable set, for example.

A big open question is always how to go to nonlinear systems. While we propose to approximate the linearized system together with the linearization error (Lagrange remainder), another option is to lift the nonlinear system via basis functions (which goes in the direction of applying Koopman operator theory). So there is not a lot of connection between our work and the second reference, but it would certainly be a very interesting idea whether Koopman theory (and DMC) can also be used to receive guaranteed reachability analysis of nonlinear systems from data (however - in my opinion - a non-trivial task).