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

TL;DR

This paper introduces a systematic method for approximating boundary layers in nonlinear singularly perturbed control systems, providing $O()$ accuracy and addressing complex boundary conditions.

Contribution

It proposes a new approach to construct approximate solutions using boundary layer functions, especially for systems with potential boundary layer occurrences.

Findings

Approximate solutions with $O()$ accuracy are developed.

The method handles nonlocal three-point boundary conditions.

The paper discusses the open controllability problem for nonlinear systems.

Abstract

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