  The Antarctic and Greenland ice sheet simulation is challenging due to
unknown parameters in the p-Stokes equations. In this work, we prove the
existence of a solution to a parameter identification for the ice rheology and
the friction coefficient. Additionally, we verify G\^ateaux differentiability
of the control-to-state operator by extending a similar result for distributed
control. Moreover, we have more complicated boundary conditions. We only have
to add a small diffusion term and assume the nonlinear exponent, which is given
in applications, to be small enough to obtain the results. Finally, we state
the adjoint equation and prove existence and uniqueness of a solution for this
equation.
