Continuous approximations of a class of piece-wise continuous systems

TL;DR

This paper develops a rigorous mathematical framework for approximating piece-wise continuous systems with continuous models, enabling better numerical analysis and control of complex dynamical systems, including fractional-order systems.

Contribution

It introduces a novel approach using differential inclusions to approximate piece-wise continuous functions, facilitating analysis and control of fractional and integer-order systems.

Findings

Provides a mathematical foundation for continuous approximations

Enables modeling of fractional-order systems with discontinuities

Demonstrates applications through examples and comparative studies

Abstract

In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise functions can be locally or globally approximated. The approximation results can be used to model piece-wise continuous-time dynamical systems of integer or fractional-order. In this way, by overcoming the lack of numerical methods for diffrential equations of fractional-order with discontinuous right-hand side, unattainable procedures for systems modeled by this kind of equations, such as chaos control, synchronization, anticontrol and many others, can be easily implemented. Several examples are presented and three comparative applications are studied.