Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects
Abstract
:1. Introduction
- Lyapunov stability: if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable.
- Asymptotic stability: if is Lyapunov stable and all solutions that start out near converge to then is said to be asymptotically stable.
- Exponential stability: this stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge.
- Probability stability: a property of probability distributions.
- Algebraic geometry stability: this stability is a notion which characterizes when a geometric object has some desirable properties for the purpose of classifying them.
- Numerical stability: a property of numerical algorithms which describes how errors in the input data propagate through the algorithm.
- K-stability: a stability condition for algebraic varieties.
- Radius stability: a property of continuous polynomial functions.
- Learning theory stability: a property of machine learning algorithms.
- (–Hilfer derivative).
- (–Riemann–Liouville integral).
- .
- (and
- (Conformable derivative),
- (Riemann–Liouville derivative).
- (Riemann–Liouville derivative).
- (Caputo derivative).
2. Preliminaries
2.1. Fox’s –Functions
- Exponential map
- Mittag-Leffler function with 1 parameter.
- Gauss Hypergeometric function
- Wright function
- –function
- Fox–Wright function
- Meijer -function
2.2. Generalized Triangular Norms (GTNs)
2.3. Fuzzy Normed Spaces
- is a linear space.
- is a family of matrix-valued fuzzy (shortly, MVF) sets .
- is a continuous increasing function.
- for all and .
- for each , and
2.4. MV Random Normed (MVRN) Spaces
- is a vector space.
- is a set of MV distribution functions (MVDFs)
- is a left-continuous and non-decreasing function.
- and
- contains functions s.t.,
- for every and
- the maximal element of is
- (1)
- iff
- (2)
- for every
- (3)
- .
2.5. Mittag-Leffler Function and Its Approximations
2.6. Fractional Derivatives
2.6.1. Hilfer Derivative
2.6.2. Conformable Derivative
2.6.3. Riemann–Liouville Derivative
2.6.4. Caputo Derivative
2.7. Fractional-Order Delayed Matrix Sine and Cosine
2.8. Aggregate Maps
- Geometric mean functions
- Arithmetric mean functions
- Maximum functions
- Minimum functions
- Median of odd numbers
- Median of even numbers
- Sum functions:
- Product functions:
2.9. Alternative Fixed Point Theory [22]
- (1)
- The fixed point of is the convergence point of ;
- (2)
- In the set , is the unique fixed point of ;
- (3)
- .
2.10. The First Kudryashov-Type Method
2.11. Gronwall Inequality
3. Fuzzy Stability Results of (1) for Case 1
3.1. Fuzzy Multistability Results for Finite Domains
3.2. Fuzzy Multi-Stability Results for Unbounded Domains
4. Results of (1) for Case 2
Application of the First Kudryashov-Type Method
5. Fuzzy Asymptotic Stability Results of (1) for Case 3
6. Fuzzy Asymptotic Stability Results of (1) for Case 4
7. Random Finite-Time Stability Results of (1) for Case 5
- (i)
- (ii)
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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(0, 0.025) | (0.025, 0.050) | (0.050, 0.075) | (0.075, 0.100) | |
---|---|---|---|---|
0.0317 | 0.0620 | 0.0829 | 0.0857 | |
0.0286 | 0.0579 | 0.0764 | 0.0810 | |
0.0214 | 0.0548 | 0.0705 | 0.0789 | |
0.0198 | 0.0413 | 0.0507 | 0.0696 | |
0.0129 | 0.0364 | 0.0489 | 0.0601 | |
0.0107 | 0.0207 | 0.0360 | 0.0549 | |
0.0095 | 0.0199 | 0.0317 | 0.0501 | |
0.0048 | 0.0076 | 0.0221 | 0.0449 |
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Aderyani, S.R.; Saadati, R.; O’Regan, D.; Alshammari, F.S. Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects. Mathematics 2024, 12, 904. https://doi.org/10.3390/math12060904
Aderyani SR, Saadati R, O’Regan D, Alshammari FS. Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects. Mathematics. 2024; 12(6):904. https://doi.org/10.3390/math12060904
Chicago/Turabian StyleAderyani, Safoura Rezaei, Reza Saadati, Donal O’Regan, and Fehaid Salem Alshammari. 2024. "Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects" Mathematics 12, no. 6: 904. https://doi.org/10.3390/math12060904
APA StyleAderyani, S. R., Saadati, R., O’Regan, D., & Alshammari, F. S. (2024). Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects. Mathematics, 12(6), 904. https://doi.org/10.3390/math12060904