  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.

Optimization and Control