  This paper studies exponential stability properties of a class of
two-dimensional (2D) systems called differential repetitive processes (DRPs).
Since a distinguishing feature of DRPs is that the problem domain is bounded in
the "time" direction, the notion of stability to be evaluated does not require
the nonlinear system defining a DRP to be stable in the typical sense. In
particular, we study a notion of exponential stability along the discrete
iteration dimension of the 2D dynamics, which requires the boundary data for
the differential pass dynamics to converge to zero as the iterations evolve.
Our main contribution is to show, under standard regularity assumptions, that
exponential stability of a DRP is equivalent to that of its linearized
dynamics. In turn, exponential stability of this linearization can be readily
verified by a spectral radius condition. The application of this result to
Picard iterations and iterative learning control (ILC) is discussed.
Theoretical findings are supported by a numerical simulation of an ILC
algorithm.
