  In this paper, we develop a novel data-driven method for Deformable Mirror
(DM) control. The developed method updates both the DM model and DM control
actions that produce desired mirror surface shapes. The novel method explicitly
takes into account actuator constraints and couples a feedback control
algorithm with an algorithm for recursive estimation of DM influence function
models. In addition to this, we explore the possibility of using Walsh basis
functions for DM control. By expressing the desired and observed mirror surface
shapes as sums of Walsh pattern matrices, we formulate the control problem in
the 2D Walsh basis domain. We thoroughly experimentally verify the developed
approach on a 140-actuator MEMS DM, developed by Boston Micromachines. Our
results show that the novel method produces the root-mean-square surface error
in the $14-40$ nanometer range. These results can additionally be improved by
tuning the control and estimation parameters. The developed approach is also
applicable to other DM types, such as for example, segmented DMs.
