Pass Profile Exponential and Asymptotic Stability of Nonlinear Repetitive Processes * *this Work Was Supported in Part by Russian Foundation for Basic Research under Grants 16-08-00916_a, 16-38-00304_mol_a and in Part by National Science Center in Poland under Grant 2015/17/B/ST7/03703.

This paper considers discrete and differential nonlinear repetitive processes using the state-space model setting. These processes are a particular class of 2D systems that have their origins in the modeling of physical processes. Their distinguishing characteristic is that one of the two independent variables needed to describe the dynamics evolves over a finite interval and therefore they are defined over a subset of the upper-right quadrant of the 2D plane. The current stability theory for nonlinear dynamics assumes that they operate over the complete upper-right quadrant and this property may be too strong for physical applications, particulary in terms of control law design. With applications in mind, the contribution of this paper is the use of vector Lyapunov functions to characterize a new properties termed pass profile exponential stability and pass profile asymptotic stability.