  This paper is concerned with optimality and stability analysis of a family of
ensemble Kalman filter (EnKF) algorithms. EnKF is commonly used as an
alternative to the Kalman filter for high-dimensional problems, where storing
the covariance matrix is computationally expensive. The algorithm consists of
an ensemble of interacting particles driven by a feedback control law. The
control law is designed such that, in the linear Gaussian setting and
asymptotic limit of infinitely many particles, the mean and covariance of the
particles follow the exact mean and covariance of the Kalman filter. The
problem of finding a control law that is exact does not have a unique solution,
reminiscent of the problem of finding a transport map between two
distributions. A unique control law can be identified by introducing control
cost functions, that are motivated by the optimal transportation problem or
Schr\"odinger bridge problem. The objective of this paper is to study the
relationship between optimality and long-term stability of a family of exact
control laws. Remarkably, the control law that is optimal in the optimal
transportation sense leads to an EnKF algorithm that is not stable.
