  We perform stability analysis of a kinetic bacterial chemotaxis model of
bacterial self-organization, assuming that bacteria respond sharply to chemical
signals. The resulting discontinuous tumbling kernel represents the key
challenge for the stability analysis as it rules out a direct linearization of
the nonlinear terms. To address this challenge we fruitfully separate the
evolution of the shape of the cellular profile from its global motion. We
provide a full nonlinear stability theorem in a perturbative setting when
chemical degradation can be neglected. With chemical degradation we prove
stability of the linearized operator. In both cases we obtain exponential
relaxation to equilibrium with an explicit rate using hypocoercivity
techniques. To apply a hypocoercivity approach in this setting, we develop two
novel and specific approaches: i) the use of the $H^1$ norm instead of the
$L^2$ norm, and ii) the treatment of nonlinear terms. This work represents an
important step forward in bacterial chemotaxis modeling from a kinetic
perspective as most results are currently only available for the macroscopic
descriptions, which are usually parabolic in nature. Significant difficulty
arises due to the lack of regularization of the kinetic transport operator as
compared to the parabolic operator in the macroscopic scaling limit.
