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Stabilization of Nonlinear Systems by Gain-Limited Feedback Laws

Bryce Christopherson, Farhad Jafari

Abstract

We study local stabilization of nonlinear control systems under explicit gain constraints on the feedback law. Using a quantitative refinement of Brockett's openness condition, we introduce the notion of a maximal continuous openness rate for the system vector field near equilibrium. Combining this with a local-section characterization of stabilizability, we derive a general necessary condition for the existence of gain-limited stabilizing feedback. This condition yields sharp no-go results for broad classes of nonlinear systems, including systems that are stabilizable only by nonsmooth feedback. Several examples illustrate how openness rates impose fundamental lower bounds on stabilizing feedback growth near an equilibrium point.

Stabilization of Nonlinear Systems by Gain-Limited Feedback Laws

Abstract

We study local stabilization of nonlinear control systems under explicit gain constraints on the feedback law. Using a quantitative refinement of Brockett's openness condition, we introduce the notion of a maximal continuous openness rate for the system vector field near equilibrium. Combining this with a local-section characterization of stabilizability, we derive a general necessary condition for the existence of gain-limited stabilizing feedback. This condition yields sharp no-go results for broad classes of nonlinear systems, including systems that are stabilizable only by nonsmooth feedback. Several examples illustrate how openness rates impose fundamental lower bounds on stabilizing feedback growth near an equilibrium point.
Paper Structure (6 sections, 11 theorems, 51 equations)

This paper contains 6 sections, 11 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Background and Technical Preliminaries
  3. The Approach
  4. Norm Estimates via Openness Rates
  5. Local Section Domain Conditions
  6. Conclusion

Key Result

Theorem 3

For $k \geq 0$, the system control sys is locally asymptotically (resp. exponentially) stabilizable by means of a $C^{k}$ feedback law $u$ if and only if there exists a $C^{k}$ local section $\alpha$ of $f$ at the origin such that $\textrm{proj}_1\circ \alpha$ is a homeomorphism (resp. diffeomorphis

Theorems & Definitions (35)

  • Definition 1: Local Asymptotic Stabilizability - Feedback Laws
  • Definition 2: Sections, Local Sections
  • Theorem 3
  • Definition 3: Locally Open
  • Theorem 4: Brockett's Theorem - Continuous Extension
  • Definition 4
  • Definition 5
  • Example 1
  • Definition 6
  • Theorem 5
  • ...and 25 more