Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Finite-time stability for uncertain differential equations: a first investigation on a new class of multi-agent systems

  • 60 Accesses

Abstract

In this paper, we discuss a new kind of stability, that is, finite-time stability, for uncertain differential equations, by formalizing some properties. As a possible application, we define a new class of uncertain multi-agent systems, according to the Liu’s uncertainty theory, as a counterpart of stochastic multi-agent systems. We formalize the governing equations, driven by canonical process, which is a type of uncertain process with stationary and independent increments. The concept of finite-time consensus in the context of uncertainty theory is consequently derived. A numerical procedure to estimate the settling time is proposed. The case with proportional delay was also considered.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Barbacioru C (2010) Uncertainty functional differential equations for finance. Surv Math Appl 5:275–284

  2. Bede B, Rudas IJ (2011) Approximation properties of fuzzy transforms. Fuzzy Sets Syst 180:20–40

  3. Brooks R, Schmitt K (2009) The contraction mapping principle and some applications. Electron J Diff Eq Monogr 9:1–90

  4. Chen X, Gao J (2013) Stability analysis of linear uncertain differential equations. Ind Eng Manag Syst 12(1):2–8

  5. Chen XW, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Mak 9(1):69–81

  6. D’Aniello G, Loia V, Orciuoli F (2015) A multi-agent fuzzy consensus model in a situation awareness framework. Appl Soft Comp 30:430–440

  7. De Falco M, Gaeta M, Loia V, Rarità L, Tomasiello S (2016) Differential quadrature based numerical solutions of a fluid dynamic model for supply chains. Commun Math Sci 14(5):1467–1476

  8. del Val E, Martanez C, Botti V (2016) Analyzing users-activity in online social networks over time through a multi-agent framework. Soft Comput 20(11):4331–4345

  9. Gao Y, Yao K (2014) Continuous dependence theorems on solutions of uncertain differential equations. Appl Math Model 38:3031–3037

  10. Gao R, Ralescu DA (2010) Uncertain wave equation for vibrating string. IEEE Trans Fuzzy Syst (in press)

  11. Hentout A, Maoudj A, Kaid-Youcef N, Hebib D, Bouzouia B (2011) Distributed multi-agent bidding-based approach for the collaborative mapping of unknown indoor environments by a homogeneous mobile robot team. J Int Sys (in press)

  12. Huang J, Liu J, Yao X (2017) A multi-agent evolutionary algorithm for software module clustering problems. Soft Comput 21(12):3415–3428

  13. Jabbarpour MR et al (2018) Applications of computational intelligence in vehicle traffic congestion problem: a survey. Soft Comput 22(7):2299–2320

  14. Ji X, Zhou J (2015) Multi-dimensional uncertain differential equation: existence and uniqueness of solution. Fuzzy Optim Decis Mak 14(4):477–491

  15. Jia L, Sheng Y (2019) Stability in distribution for uncertain delay differential equation. Appl Math Comput 343:49–56

  16. Li P, Xu S, Chen W, Wei Y, Zhang Z (2018) Adaptive finite-time flocking for uncertain nonlinear multi-agent systems with connectivity preservation. Neurocomputing 275:1903–1910

  17. Liu B (2007) Uncertainty theory. Springer, Berlin

  18. Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3–16

  19. Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10

  20. Liu K, Ji Z (2017) Consensus of multi-agent systems with time delay based on periodic sample and event hybrid control. Neurocomputing 270:11–17

  21. Loia V, Tomasiello S, Vaccaro A (2017) Using fuzzy transform in multi-agent based monitoring of smart grids. Inf Sci 388(389):209–224

  22. Miao G, Ma Q, Liu Q (2016) Consensus problems for multi-agent systems with nonlinear algorithms. Neural Comput Appl 27(5):1327–1336

  23. Ming P et al (2016) Consensus stabilization in stochastic multi-agent systems with Markovian switching topology, noises and delay. Neurocomputing 200:1–10

  24. Perfilieva I (2006) Fuzzy transforms: theory and applications. Fuzzy Sets Syst 157:993–1023

  25. Qin J, Fu W, Gao H, Zheng WX (2017) Distributed k-means algorithm and fuzzy c-means algorithm for sensor networks based on multiagent consensus theory. IEEE Trans Cybern 47(3):772–783

  26. Sheng Y, Gao J (2016) Exponential stability of uncertain differential equation. Soft Comput 20(9):3673–3678

  27. Song Y-Z (2016) Consensus of agents with mixed linear discrete dynamics. Int J Control Autom Syst 14(4):1139–1143

  28. Su T, Wu H, Zhou J (2016) Stability of multi-dimensional uncertain differential equation. Soft Comput 20(12):4991–4998

  29. Tomasiello S (2011) A note on three numerical procedures to solve Volterra integro-differential equations in structural analysis. Comput Math Appl 62:3183–3193

  30. Tomasiello S (2017) An alternative use of fuzzy transform with application to a class of delay differential equations. Int J Comput Math 94(9):1719–1726

  31. Tomasiello S, Macias-Diaz JE (2017) Note on a picard-like method for caputo fuzzy fractional differential equations. Appl Math Inf Sci 11(1):281–287

  32. Tomasiello S, Gaeta M, Loia V (2016) Quasi-consensus in second-order multi-agent systems with sampled data through fuzzy transform. J Uncertain Syst 10(4):3–10

  33. Victor P, Tsigaridas E, Zhao L (2015) Simple and efficient real root-finding for a univariate polynomial (to appear). arxiv.org

  34. Wang X, Ning Y (2017) Stability of uncertain delay differential equations. J Intell Fuzzy Syst 32:2655–2664

  35. Yang X, Gao J (2016) Linear-quadratic uncertain differential game with application to resource extraction problem. IEEE Trans Fuzzy Syst 24(4):819–826

  36. Yang X, Ni Y, Zhang Y (2017) Stability in inverse distribution for uncertain differential equations. J Intell Fuzzy Syst 32(3):20512059

  37. Yao K, Chen X (2013) A numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825–832

  38. Yin J, Khoo S, Man Z, Yu X (2011) Finite-time stability and instability of stochastic nonlinear systems. Automatica 47(12):2671–2677

  39. Zhang Y, Gao J, Huang Z (2017) Hamming method for solving uncertain differential equations. Appl Math Comput 313:331–341

  40. Zhao L, Jia Y (2015) Finite-time consensus for second-order stochastic multi-agent systems with nonlinear dynamics. Appl Math Comput 270:278–290

  41. Zheng Y, Chen W, Wang L (2011) Finite-time consensus for stochastic multi-agent systems. Int J Control 84(10):1644–1652

Download references

Acknowledgements

Mrs Neda Gossili acknowledges the support from the Iranian Ministry of University and Research for spending six months of her PhD programme at the University of Salerno under the supervision of Dr Stefania Tomasiello. Mr Santiago Marín Mejía acknowledges the support from Maestría en Ingeniería de Sistemas y Computacion, Universidad Tecnologica de Pereira.

Author information

Correspondence to S. Tomasiello.

Ethics declarations

Conflicts of interest

The authors declare that they have no potential conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by V. Loia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tomasiello, S., Marín Mejía, S. & Gossili, N. Finite-time stability for uncertain differential equations: a first investigation on a new class of multi-agent systems. Soft Comput 24, 3275–3284 (2020). https://doi.org/10.1007/s00500-019-04086-0

Download citation

Keywords

  • Uncertain differential equation
  • Liu’s process
  • Stability
  • Consensus
  • Fuzzy transform