# A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems

## TL;DR

This paper introduces a variational approach based on linear openness and transversality properties to achieve local exponential stabilization of nonlinear systems, extending classical criteria and handling systems with higher-order stabilization terms.

## Contribution

It develops a novel variational framework that generalizes existing stabilization conditions, allowing exponential stabilization under weaker rank conditions and higher-order effects.

## Key findings

- Linear openness and transversality suffice for local exponential stabilization.
- The approach extends classical Kalman and Hautus criteria.
- It handles systems stabilized through higher-order terms.

## Abstract

Local asymptotic stabilizability is a topic of great theoretical interest and practical importance. Broadly, if a system $\dot{x} = f(x,u)$ is locally asymptotically stabilizable, we are guaranteed a feedback controller $u(x)$ that forces convergence to an equilibrium for trajectories initialized sufficiently close to it. A necessary condition was given by Brockett: such controllers exist only when $f$ is open at the equilibrium. Recently, Gupta, Jafari, Kipka and Mordukhovich considered a modification to this condition, replacing Brockett's topological openness by the linear openness property of modern variational analysis. In this paper, we show that under the linear openness assumption there is a local diffeomorphism of neighborhoods of the equilibirum on which the system is exponentially stabilizable by means of continuous stationary feedback laws. Introducing a transversality property and relating it to the above diffeomorphism, we prove that linear openness and transversality on a punctured neighborhood of an equilibrium is sufficient for local exponential stabilizability of systems with a rank deficient linearization.The main result goes beyond the usual Kalman and Hautus criteria for the existence of exponential stabilizing feedback laws, since it allows us to handle systems for which exponential stabilization is achieved through higher-order terms. However, it is implemented so far under a rather restrictive row-rank conditions. This suggests a twofold approach to the use of these properties: a point-wise version is enough to ensure stability via the linearization, while a local version is enough to overcome deficiencies in the linearization.