1. Z. Liu, N. Ozay, and E. D. Sontag. Properties of immersions for systems with multiple limit sets with implications to learning Koopman embeddings. Automatica, 176:112226, 2025. [PDF] Keyword(s): linear systems, nonlinear systems, observables, Koopman embedding, duality. Abstract:

   Linear immersions (or Koopman eigenmappings) of a nonlinear system have wide applications in prediction and control. In this work, we study the non-existence of one-to-one linear immersions for nonlinear systems with multiple omega-limit sets. While previous research has indicated the possibility of discontinuous one-to-one linear immersions for such systems, it remained uncertain whether continuous one-to-one linear immersions are attainable. Under mild conditions, we prove that any continuous one-to-one immersion to a class of systems including linear systems cannot distinguish different omega-limit sets, and thus cannot be one-to-one. Furthermore, we show that this property is also shared by approximate linear immersions learned from data as sample size increases and sampling interval decreases. Multiple examples are studied to illustrate our results.




2. Y. Wang and E.D. Sontag. On two definitions of observation spaces. Systems Control Lett., 13(4):279-289, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90116-3] Keyword(s): observability, identifiability, observables, observation space, realization theory. Abstract:

   This paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls.

1. Z. Liu, N. Ozay, and E. D. Sontag. On the non-existence of immersions for systems with multiple omega-limit sets. In 22nd IFAC World Congress, IFAC-PapersOnLine, volume 56, pages 60-64, 2023. Note: This is a preliminary version of the journal paper Properties of immersions for systems with multiple limit sets with implications to learning Koopman embeddings.[PDF] [doi:https://doi.org/10.1016/j.ifacol.2023.10.1408] Keyword(s): linear systems, nonlinear systems, observables, Koopman embedding, duality. Abstract:

   Linear immersions (or Koopman eigenmappings) of a nonlinear system have wide applications in prediction and control. In this work, we study the existence of one-to-one linear immersions for nonlinear systems with multiple omega-limit sets. For this class of systems, existing work shows that a discontinuous one-to-one linear immersion may exist, but it is unclear if a continuous one-to-one linear immersion exists. Under mild conditions, we prove that systems with multiple omega-limit sets cannot admit a continuous one-to-one immersion to a class of systems including linear systems.




2. E.D. Sontag. Spaces of observables in nonlinear control. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel, pages 1532-1545, 1995. Birkhäuser. [PDF] Keyword(s): observability, dynamical systems. Abstract:

   Invited talk at the 1994 ICM. Paper deals with the notion of observables for nonlinear systems, and their role in realization theory, minimality, and several control and path planning questions.




3. Y. Wang and E.D. Sontag. A new result on the relation between differential-algebraic realizability and state space realizations. In Proc. Conf. Info. Sciences and Systems, Johns Hopkins University Press, 1989, pages 143-147, 1989. Keyword(s): observables, observation space, identifiability, observability, realization theory.

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