  In this paper, we study the global controllability and stabilization problems
of the harmonic map heat flow from a circle to a sphere. Combining ideas from
control theory, heat flow, differential geometry, and asymptotic analysis, we
obtain several important properties, such as small-time local controllability,
local quantitative rapid stabilization, obstruction to semi-global asymptotic
stabilization, and global controllability to geodesics. Surprisingly, due to
the geometric feature of the equation we also discover the small-time global
controllability between harmonic maps within the same homotopy class for
general compact Riemannian manifold targets, which is to be compared with the
analogous but longstanding problem for the nonlinear heat equations.
