  In this paper, the convergence of an algorithm for recovering the unknown
kinematic viscosity of a two-dimensional incompressible, viscous fluid is
studied. The algorithm of interest is a recursive feedback control-based
algorithm that leverages observations that are received continuously-in-time,
then dynamically provides updated values of the viscosity at judicious moments.
It is shown that in an idealized setup, convergence to the true value of the
viscosity can indeed be achieved under a natural and practically verifiable
non-degeneracy condition. This appears to be first such result of its kind for
parameter estimation of nonlinear partial differential equations. Analysis for
two parameter update rules is carried out: one which involves instantaneous
evaluation in time and the other, averaging in time. The proofs of convergence
for either rule exploits sensitivity-type bounds in higher-order Sobolev
topologies, while the instantaneous version particularly requires delicate
energy estimates involving the time-derivative of the sensitivity-type
variable. Indeed, a crucial component in the analysis of the first update rule
is the identification of a dissipative structure for the time-derivative of the
sensitivity-type variable, which ultimately ensures a favorable dependence on
the tuning parameter of the algorithm.
