Solutions of a Class of Switch Dynamical Systems

Area of Science:

• Dynamical Systems and Control Theory
• Mathematical Analysis
• Applied Mathematics

Background:

• Investigating solutions for switch dynamical systems with discontinuous right-hand sides.
• Discontinuities often modeled by signum or Heaviside functions.
• Challenges in analyzing and numerically integrating such systems.

Purpose of the Study:

• To present a novel approach for solving discontinuous differential equations.
• To transform discontinuous systems into continuous, single-valued differential equations.
• To establish conditions for the existence and uniqueness of solutions.

Main Methods:

• Restating the initial value problem as a differential inclusion via Filippov's regularization.
• Transforming the differential inclusion into a continuous, single-valued differential equation using approximate selection results.
• Introducing the strengthened one-sided Lipschitz Condition for uniqueness.

Main Results:

• A novel method for analyzing solutions of discontinuous switch dynamical systems.
• Demonstration of existence and a sufficient uniqueness condition.
• Highlighting potential numerical integration errors when discontinuities are ignored, using examples like a preloaded compliance system.

Conclusions:

• The proposed method effectively handles discontinuous dynamical systems.
• The strengthened one-sided Lipschitz Condition provides a robust uniqueness criterion.
• Accurate numerical integration requires explicit consideration of system discontinuities.