  In this paper the finite-time stabilization problem is solved for a linear
time-varying system with unknown control direction by exploiting a modified
version of the classical extremum seeking algorithm. We propose to use a
suitable oscillatory input to modify the system dynamics, at least in an
average sense, so as to satisfy a Differential Linear Matrix Inequality (DLMI)
condition which in turns guarantees that the system's state remains inside a
prescribed time varying hyper-ellipsoid in the state space. The finite-time
stability (FTS) of the averaged dynamics implies the FTS of the original
system, as the distance between the original and the averaged dynamics can be
made arbitrarily small by choosing a sufficiently high value of the dithering
frequency used by the extremum seeking algorithm. An estimate of the necessary
minimum dithering/mixing frequency is provided, and the effectiveness of the
proposed finite-time stabilization approach is analysed by means of numerical
examples.
