# Partial Stability Concept in Extremum Seeking Problems

## TL;DR

This paper investigates extremum seeking control for systems where the cost depends only on some state variables, introducing partial stability concepts and providing conditions for practical stability using Lyapunov and averaging methods.

## Contribution

It introduces a new partial stability framework for extremum seeking problems and derives sufficient conditions using Lie bracket approximations.

## Key findings

- Conditions for practical partial stability are established.
- Broad class of extremum-seeking controllers ensuring partial stability is described.
- Theoretical results are demonstrated on Brockett integrator and rigid body examples.

## Abstract

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov's direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system with oscillating inputs are derived with the use of Lie bracket approximation techniques. These conditions are exploited to describe a broad class of extremum-seeking controllers ensuring the partial stability of the set of minima of a cost function. The obtained theoretical results are illustrated by the Brockett integrator and rotating rigid body.