Impulsive stabilization of fractional differential systems.
Liguang Xu1, Jiankang Li2, Shuzhi Sam Ge3
1Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, PR China; Department of Electrical and Computer Engineering, National University of Singapore, 117576, Singapore.
ISA Transactions
|June 24, 2017
Summary
Impulsive control can stabilize unstable fractional differential systems (FDSs). This study establishes criteria for global exponential stability and boundedness in FDSs, demonstrating control effectiveness through simulations.
Area of Science:
- Control Theory
- Fractional Calculus
- Dynamical Systems
Background:
- Fractional differential systems (FDSs) are increasingly studied for modeling complex phenomena.
- Understanding the stability and boundedness of FDSs is crucial for their practical applications.
- Impulsive effects are common in real-world systems and significantly impact dynamics.
Purpose of the Study:
- To investigate the impulsive stabilization problem for fractional differential systems (FDSs).
- To establish criteria for achieving global exponential stability and ultimate boundedness in FDSs under impulsive control.
- To demonstrate the effectiveness of impulsive control in stabilizing and bounding otherwise unstable FDSs.
Main Methods:
- Utilizing Lyapunov functions to analyze system stability.
- Employing algebraic inequality techniques for deriving control conditions.
- Leveraging the boundedness properties of Mittag-Leffler functions relevant to FDSs.
Main Results:
- Established criteria for global exponential stability of impulsive FDSs.
- Developed criteria for the ultimate boundedness of solutions for impulsive FDSs.
- Demonstrated that unstable and unbounded FDSs can be rendered stable and bounded via impulsive control.
Conclusions:
- Impulsive control is an effective strategy for stabilizing and bounding fractional differential systems.
- The theoretical results provide a robust framework for designing controllers for FDSs.
- Simulations confirm the practical applicability and effectiveness of the proposed impulsive stabilization methods.
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