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My question is based on thoughts after reading to a specific section in the paper "On Contraction Analysis for Nonlinear Systems" by W. Lohmiller and JJ. Slotine, Section 4.2 Constrained Systems. Those interested in answering my question should probably read this section first.

Question

Suppose I have an $n$ dimensional nonlinear dynamical system $$x'(t)=f(x) \quad x\in \mathbb{R}^n $$ with converges to a $m$-dimensional submanifold $\mathcal{M}=\{x:g(x)=0\}$ for any initial condition ($\mathcal{M}$ is not invariant for all $t$, but all attracting invariant subsets and equilibrium points of the system are in $\mathcal{M}$. I am mentioning this specifically because the background I have I read on constrained systems assumes $\mathcal{M}$ is invariant which is not the case here). Can I rewrite the system in terms of $n-1$ dimensional ODE with a constraint, such that the ode system has the same dynamics, is of the form of Section 4.2? That is $$z'=\underbrace{f(z)}_{\text{constrained dynamics $z=g(x)$}}+\underbrace{n}_{\text{Superimposed flow normal to the manifold}} $$

Background to my question

After reading a review on numerical methods for Lypunov functions and "On Contraction Analysis for Nonlinear Systems" I am in love with using contraction metrics for analysis of dynamical systems as it helps me answer some complex problems (none are related directly to control theory). I have a system that has the properties specified in the question. If I could apply the results as seen in Section 4.2 to my system (separating the flow projected onto a manifold and flow normal to a manifold) that would be great.

Notes:

  • If you need any clarification please comment.
  • If it is possible to separate a dynamical system in terms of flow normal to a manifold and flow projected onto a manifold that would be acceptable as well, as that is essentially what I am looking for.
  • I am not familiar with constrained dynamical systems. I may be missing something quite obvious. Feel free to tell me, so I can update my post as necessary.
  • If there are publications that specifically expand/apply the results of Section 4.2 I would greatly appreciate any citations.
  • Feel free to edit the tags of my question. I have put in the control tag given the material I have read and the rest of the content of the paper I thought it was appropriate.
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