On the approximation of the boundary layers for the controllability problem of nonlinear singularly perturbed systems

摘要

A new systematic approach to the construction of approximate solutions to aclass of nonlinear singularly perturbed feedback control systems using theboundary layer functions especially with regard to the possible occurrence ofthe boundary layers is proposed. For example, problems with feedback control,such as the steady-states of the thermostats, where the controllers add orremove heat, depending upon the temperature registered in another place of theheated bar, can be interpreted with a second-order ordinary differentialequation subject to a nonlocal three--point boundary condition. The$O(\epsilon)$ accurate approximation of behavior of these nonlinear systems interms of the exponentially small boundary layer functions is given. At the endof this paper, we formulate the unsolved controllability problem for nonlinearsystems.