Mittag-Leffler string stability of singularly perturbed stochastic systems within local fractal space
In this paper, we define a new type of string stability based on Mittag-Leffler function so-called Mittag-Leffler (p\alpha)-string stability. This kind of stability for a class of singularly perturbed stochastic systems of fractional order will be considered. The fractional derivative in these systems is in the local sense. String stability indicates uniform boundedness of the interconnected system if the initial cases of interconnected system be uniformly bounded. The deduction of the sufficient conditions of stability is based on a mixture of the concept of the Mittag-Leffler stability with the notion of p-mean string stability of singularly perturbed stochastic systems. In the sequel, our purpose is to investigate the full order system in their lower order subsystems, i.e., the reduced order system and the boundary layer correction.
This work is licensed under a Creative Commons Attribution 4.0 International License.
D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo. Fractional calculus models and numerical methods. World Scientific, 2012. https://doi.org/10.1142/8180.
H. Delavari, D. Baleanu and J. Sadati. Stability analysis of Caputo fractionalorder nonlinear systems revisited. Nonlinear Dynam., 67:2433–2439, 2012. https://doi.org/10.1007/s11071-011-0157-5.
J.L. Doob. Stochastic processes. Wiley Chapman and Hall, New York, 1953.
E.B. Dynkin. Markov processes. In Fizmatgiz, Moscow, 1963, English transl. Die Grundlehren der Math., p. 122, Wissenschaften, Bande 121, 1965. Academic Press, New York.
M. El-Ansary. Stochastic feedback design for a class of nonlinear singularly perturbed systems. Int. J. Syst. Sci., 22:2013–2023, 1991. https://doi.org/10.1080/00207729108910767.
M. El-Ansary and H.K. Khalil. On the interplay of singular perturbations and wide-band stochastic fluctuations. SIAM J. Control Optim., 24:83–94, 1986. https://doi.org/10.1137/0324004.
S.A. El-Shehawy. On properties of fractional probability measure. Int. Math. Forum, 11:1175–1184, 2016. https://doi.org/10.12988/imf.2016.610132.
I.I. Gikhman and A.V. Skorokhod. Introduction to the theory of random processes. In Nauka, Moscow 1965, English transl. Saunders, Philadelphia, 1969.
R. Gorenflo, A. Kilbas, F. Mainardi and S. Rogosin. Mittag-Leffler functions, related topics and applications. Springer-Verlag, Berlin-Heidelberg, 2014.
H.J. Haubold, A.M. Mathai and R.K. Saxena. Mittag-Leffler functions and their applications. arXiv:0909.0230v2 [math.CA], 4:1–51, 2009.
R.Z. Khasminski. Stochastic stability of differential equations. Groningen, The Netherlands: Sijthoff and Noordhoff, 1980. https://doi.org/10.1007/978-94-009- 9121-7.
A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations. Elsevier, Amsterdam, 2006.
K.M. Kolwankar and A.D. Gangal. Fractional differentiability of nowhere differentiable functions and dimensions. Chaos, 6(4):505–513, 1996. https://doi.org/10.1063/1.166197.
Y. Li, Y.Q. Chen and I. Podlubny. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl., 59:1810–1821, 2010. https://doi.org/10.1016/j.camwa.2009.08.019.
M. Liao, X.J. Yang and Q. Yan. A new viewpoint to Fourier analysis in fractal space. Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-6393- 1 26.
M. Loeve. Probability theory. Foundations. Random sequences. Van Nostrand, Princeton, 1960.
G. Mittag-Leffler. Sur la nouvelle fonction e(x). C. R. Acad. Sci. Paris, 137:554– 558, 1903.
H. Mostafaei and P.A. Ghotbi. Fractional probability measure and its properties. J. Sci. Islamic Rep. Iran, 21:259–264, 2010.
V.S. Pugachev. The theory of random functions and its application to control problems. In Fizmatgiz, Moscow, 1960, volume 5 of English transl. Internat. Series of Monographs on Automation and Automatic Control. Pergamon Press, Oxford, Addison-Wesley, 1965.
L. Rybarska-Rusinek. String stability of singularly perturbed stochastic systems. Ann. Math. Sil., 16:43–55, 2003.
M. Zeki Sarikaya and H. Budak. Generalized Ostrowski type inequalities for local fractional integrals. P. Am. Math. Soc., 145(4):1527–1538, 2015.
K. Sayevand and K. Pichaghchi. Successive approximation: A survey on stable manifold of fractional differential systems. Fract. Calculus Appl. Anal., 18:621– 641, 2015. https://doi.org/10.1515/fca-2015-0038.
M.V. Simkin and V.P. Roychowdhury. Stochastic modeling of a serial killer. J. Theor. Biol., 355:111–116, 2014. https://doi.org/10.1016/j.jtbi.2014.03.039.
L. Socha. Stochastic stability of interconnected string systems. Chaos, Soliton. Fract., 19:949–955, 2004. https://doi.org/10.1016/S0960-0779(03)00266-2.
D. Swaroop and J.K. Hedrick. String stability of interconnected systems. IEEE Trans. Autom. Control, 41:349–356, 1996. https://doi.org/10.1109/9.486636.
B.J. West. A fractional probability calculus view of Allometry. Systems, 2:89– 118, 2014. https://doi.org/10.3390/systems2020089.
X.J. Yang. Local fractional functional analysis and its applications. Asian Academic Publisher, Hong Kong, 2011.
X.J. Yang. Local fractional integral transforms. Prog. Nonlinear Scie., 4:1–225, 2011.
X.J. Yang. Advanced local fractional calculus and its applications. World Science Publisher, New York, 2012.
X.J. Yang, D. Baleanu and H.M. Srivastava. Local fractional integral transforms and their applications. Academic Press, New York, 2015.
X.J. Yang, F. Gao, J.A. Tenreiro Machado and D. Baleanu. Exact travelling wave solutions for local fractional partial differential equations in mathematical physics. Math. Methods Eng., 24:175–191, 2019. https://doi.org/10.1007/978-3- 319-90972-1 12.
X.J. Yang, F. Gao and H.M. Srivastava. Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations. Compu. Math. Appl., 73(2):203–210, 2017. https://doi.org/10.1016/j.camwa.2016.11.012.
X.J. Yang, F. Gao and H.M. Srivastava. A new computational approach for solving nonlinear local fractional PDEs. J. Comput. Appl. Math., 339:285–296, 2018. https://doi.org/10.1016/j.cam.2017.10.007.