ODEs with Constraints

[EthanJamesLew]

I have been going through the benchmarks from the original Arch competition, and some of them have constraints that are not implemented. For example: “Prde20” has a constraint that: x(t) + y(t) + z(t) = 10 for all t. Does Autokoopman currently have support for implementing these constraints for continuous systems? If not, could we add some support for constraints on dynamics?

[EthanJamesLew]

@Abdu-Hekal Our implementation of Prde20 should be valid if the initial conditions satisfy the equality $$x_0 + y_0 + z_0 = 10$$. This can be expressed as the DAE

$$ \begin{cases} \dot x = - \frac{x y}{x + 1} \ \dot y = \frac{x y}{x + 1} - \alpha y \ 10 = x + y + z \ \end{cases} $$

but it should be sufficient as is.

At any rate, I'm adding an implementation of DAEs, but this shouldn't be a blocker for the benchmarks.

[Abdu-Hekal]

@EthanJamesLew I believe for the prde20 benchmark, the condition must hold for all times t (shown in the image attached) and not only as an initial condition.

If we can express the DAE as you outlined: 10 = x + y + z, then we can capture this condition correctly.

Thank you!

[EthanJamesLew]

@Abdu-Hekal the evolution function should already enforce $$x(t) + y(t) + z(t) = c_0, \quad \forall t \in \mathbb R^+$$ where $$x_0 + y_0 + z_0 = c_0$$. The DAE complains about being overconstrained when having the 3 differential equations + 1 constraint. Can you find an initial condition where that's not the case?

[Abdu-Hekal]

@EthanJamesLew You are correct! I did not notice that the evolution function implicitly ensures that this condition is always satisfied. This also seems to be the case for the robe21 benchmark. It seems we don't need support for these constraints for the sake of these benchmarks after all. Thank you!