Abstract
This project studies polynomial observables that are compatible with a lifted linear system and define invariant graphs of the form p = p(x). Such observables arise naturally in the context of Koopman theory and super-linearization, where one seeks finite-dimensional linear representations of nonlinear dynamics.
Starting from the invariance identity, we derive algebraic conditions that constrain the structure of admissible polynomials. In the two-dimensional case, a complete analysis shows that all nonlinear terms are forced to depend on a single direction determined by the control vector B. This leads to a canonical representation in which the observable takes the form of a univariate polynomial in a linear functional v^\top x.
We then extend this analysis to general degree k, obtaining explicit compatibility conditions on the system matrices and coefficients. A dichotomy appears depending on a scalar parameter: in the generic case only the highest-degree term survives, while in the degenerate case lower-order terms are allowed.
Finally, we investigate the higher-dimensional setting. By introducing a subspace defined by the annihilation of higher-order gradients, we obtain an invariant decomposition that leads to a block-triangular form of the system. In the diagonal case, the problem can also be interpreted as a first-order linear partial differential equation, whose solutions can be characterized using characteristic curves and lattice constraints.
Document Type
Article
Class Name
Electrical and Systems Engineering Undergraduate Research
Language
English (en)
Date of Submission
4-22-2026
Recommended Citation
Tian, Chengcheng, "Polynomial Invariant Observables and Finite-Dimensional Linear Representations" (2026). Electrical and Systems Engineering Undergraduate and Graduate Research. 56.
https://openscholarship.wustl.edu/eseundergraduate_research/56