On Finite/Fixed-Time Stability Theorems of Discontinuous Differential Equations
Abstract
:1. Introduction
2. Materials and Methods
- (i)
- FNTS, if it satisfies Lyapunov stability and finite-time attractiveness. Moreover, the finite-time attractiveness means that there exists a satisfying and for all , where is known as the ST.
- (ii)
- Uniformly FNTS, if it has uniform Lyapunov stability and finite-time attractiveness.
- (iii)
- Uniformly FXTS, if it is uniformly FNTS and is uniformly bounded with respect to , i.e., there exists a fixed constant for any initial state point values satisfying .
- (1)
- is positive definite,
- (2)
- is radially unbounded and
- (3)
- there exist , , such that
- (1)
- is positive definite,
- (2)
- is radially unbounded and
- (3)
- there exist , and , such that
3. Results
- (H1):
- (H2):
- , where ;
- (H3):
- there exists satisfying
- , where ;
- there exists such that
- : , where , .
4. Numerical Simulations
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
FNTS | finite time stability; |
FXTS | fixed time stability; |
DDE | discontinuous differential equation; |
LF | Lyapunov function; |
ST | settling time; |
USC | upper semi-continuous; |
DI | differential inclusion; |
LLC | locally Lipschitz continuous. |
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Li, L.; Wang, D. On Finite/Fixed-Time Stability Theorems of Discontinuous Differential Equations. Mathematics 2022, 10, 2221. https://doi.org/10.3390/math10132221
Li L, Wang D. On Finite/Fixed-Time Stability Theorems of Discontinuous Differential Equations. Mathematics. 2022; 10(13):2221. https://doi.org/10.3390/math10132221
Chicago/Turabian StyleLi, Luke, and Dongshu Wang. 2022. "On Finite/Fixed-Time Stability Theorems of Discontinuous Differential Equations" Mathematics 10, no. 13: 2221. https://doi.org/10.3390/math10132221
APA StyleLi, L., & Wang, D. (2022). On Finite/Fixed-Time Stability Theorems of Discontinuous Differential Equations. Mathematics, 10(13), 2221. https://doi.org/10.3390/math10132221