Quantify openness
Use a variational property
Linear openness strengthens Brockett’s topological condition with a scale-sensitive local covering estimate.
Paper project · Nonlinear control
A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems
Overview
Higher-order terms can stabilize what the linearization misses.
Classical linear tests can fail when the linearization of a nonlinear control system is rank deficient, even though higher-order terms still contain stabilizing information. The paper uses linear openness from variational analysis to formulate sufficient conditions beyond those tests.
Under linear openness, a local change of coordinates yields exponential stabilizability by continuous stationary feedback. With an additional transversality condition on a punctured neighborhood, the result also handles certain rank-deficient linearizations. The theorem is deliberately qualified by the row-rank assumptions under which the construction is proved.
Add transversality
Control behavior off the equilibrium
A punctured-neighborhood condition helps overcome deficiencies that are invisible to the rank-deficient linearization alone.
Recover exponential decay
Exploit nonlinear terms
The resulting sufficient conditions reach systems whose stabilization depends on higher-order structure, subject to the theorem’s row-rank hypotheses.
Citation
Use or reference this work.
The published article and an open arXiv version are linked above.
BibTeX citation
@article{christopherson2020variational,
title = {A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems},
author = {Christopherson, Bryce A. and Jafari, Farhad and Mordukhovich, Boris S.},
journal = {SN Operations Research Forum},
volume = {1},
eid = {5},
pages = {1--27},
year = {2020},
doi = {10.1007/s43069-020-0003-z}
}