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