American Institute of Mathematical Sciences

Advanced
April  2020, 25(4): 1439-1467. doi: 10.3934/dcdsb.2019235

Periodicity and stabilization control of the delayed Filippov system with perturbation

Zuowei Cai a,b,c, , Jianhua Huang c,, , Liu Yang a, and  Lihong Huang d,

a. 

Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang, Hunan 421002, China
b. 

School of Information Science and Engineering, Hunan Women's University, Changsha, Hunan 410002, China
c. 

College of Science, National University of Defense Technology, Changsha, Hunan 410073, China
d. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China

* Corresponding author: Jianhua Huang

Received  August 2018 Revised  May 2019 Published  April 2020 Early access  November 2019

Fund Project: The first author is supported by National Natural Science Foundation of China (No.11701172, No.11771449), China Postdoctoral Science Foundation (No.2017M613361), the Science and Technology Plan Project of Hunan Province (No.2016TP1020), Open fund project of Hunan Provincial Key Laboratory of Intelligent Information Processing and Application for Hengyang normal university(No.2017IIPAZD02), Teaching Reform Project of Ordinary Colleges and Universities in Hunan Province (No. 911).

By employing Leray-Schauder alternative theorem of set-valued maps and non-Lyapunov method (non-smooth analysis, inequality analysis, matrix theory), this paper investigates the problems of periodicity and stabilization for time-delayed Filippov system with perturbation. Several criteria are obtained to ensure the existence of periodic solution of time-delayed Filippov system by using differential inclusion. By designing appropriate switching state-feedback controller, the asymptotic stabilization and exponential stabilization control of Filippov system are realized. Applying these criteria and control design method to a class of time-delayed neural networks with perturbation and discontinuous activation functions under a periodic environment. The developed theorems improve the existing results and their effectiveness are demonstrated by numerical example.

Keywords: Time-delay, Filippov system, differential inclusion, periodic solution, stabilization.
Mathematics Subject Classification: Primary: 39A23, 34D20, 34K09; Secondary: 34K20.
Citation: Zuowei Cai, Jianhua Huang, Liu Yang, Lihong Huang. Periodicity and stabilization control of the delayed Filippov system with perturbation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1439-1467. doi: 10.3934/dcdsb.2019235
References:
[1]

J.-P. Aubin and A. Cellina, Differential Inclusions, Set-Valued Functions and Viability Theory, Grundlehren der Mathematischen Wissenschaften, 264. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[3]

M. Benchohra and S. K. Ntouyas, Existence results for functional differential inclusions, E. J. Diff. Equ., (2001), 8 pp.   Google Scholar

[4]

A. Boucherif and C. C. Tisdell, Existence of periodic and non-periodic solutions to systems of boundary value problems for first-order differential inclusions with super-linear growth, Appl. Math. Comput., 204 (2008), 441-449.  doi: 10.1016/j.amc.2008.07.001.  Google Scholar

[5]

Z. W. Cai, J. H. Huang and L. H. Huang, Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 3591-3614.  doi: 10.3934/dcdsb.2017181.  Google Scholar

[6]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[7]

J. Cortés, Discontinuous dynamical systems: A tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Syst. Mag., 28 (2008), 36-73.  doi: 10.1109/MCS.2008.919306.  Google Scholar

[8]

F. S. De Blasi, L. Górniewicz and G. Pianigiani, Topological degree and periodic solutions of differential inclusions, Nonlinear Anal., 37 (1999), 217-245.  doi: 10.1016/S0362-546X(98)00044-3.  Google Scholar

[9]

B. C. Dhage, Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications, Comput. Math. Appl., 51 (2006), 589-604.  doi: 10.1016/j.camwa.2005.07.017.  Google Scholar

[10]

J. Dugundji and A. Granas, Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.  Google Scholar

[11]

A. F. Filippov, Differential Equations with Discontinuous Right-hand Side, Mathematics and Its Applications (Soviet Series), 18. Kluwer Academic, Boston, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[12]

M. Forti, M. Grazzini, P. Nistri and L. Pancioni, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Phys. D, 214 (2006), 88-99.  doi: 10.1016/j.physd.2005.12.006.  Google Scholar

[13]

G. Haddad, Monotone viable trajectories for functional differential inclusions, J. Differential Equations, 42 (1981), 1-24.  doi: 10.1016/0022-0396(81)90031-0.  Google Scholar

[14]

G. Haddad, Topological properties of the sets of solutions for functional differential inclusions, Nonlinear Anal., 5 (1981), 1349-1366.  doi: 10.1016/0362-546X(81)90111-5.  Google Scholar

[15]

J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[16]

H. Hermes, Discontinuous Vector Fields And Feedback Control, Differential Equations and Dynamical Systems, Academic, New York, (1967), 155–165.  Google Scholar

[17]

S. H. Hong, Existence results for functional differential inclusions with infinite delay, Acta Math. Sin., 22 (2006), 773-780.  doi: 10.1007/s10114-005-0600-y.  Google Scholar

[18]

S. H. Hong and L. Wang, Existence of solutions for integral inclusions, J. Math. Anal. Appl., 317 (2006), 429-441.  doi: 10.1016/j.jmaa.2006.01.057.  Google Scholar

[19]

S. C. Hu and N. S. Papageorgiou, On the existence of periodic solutions for nonconvex valued differential inclusions in $R^{N}$, Proc. Amer. Math. Soc., 123 (1995), 3043-3050.  doi: 10.2307/2160658.  Google Scholar

[20]

S. C. Hu, D. A. Kandilakis and N. S. Papageorgiou, Periodic solutions for nonconvex differential inclusions, Proc. Amer. Math. Soc., 127 (1999), 89-94.  doi: 10.1090/S0002-9939-99-04338-5.  Google Scholar

[21]

J. P. Lasalle, The Stability of Dynamical System, SIAM, Philadelphia, 1976.  Google Scholar

[22]

A. Lasota and Z. Opial, An application of the Kukutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys., 13 (1965), 781-786.   Google Scholar

[23]

Y. Li and Z. Lin, Periodic solutions of differential inclusions, Nonlinear Anal., 24 (1995), 631-641.  doi: 10.1016/0362-546X(94)00111-T.  Google Scholar

[24]

G. C. Li and X. P. Xue, On the existence of periodic solutions for differential inclusions, J. Math. Anal. Appl., 276 (2002), 168-183.  doi: 10.1016/S0022-247X(02)00397-9.  Google Scholar

[25]

K.-Z. Liu, X.-M. Sun, J. Liu and A. R. Teel, Stability theorems for delay differential inclusions, IEEE Trans. Autom. Control, 61 (2016), 3215-3220.  doi: 10.1109/TAC.2015.2507782.  Google Scholar

[26]

V. Lupulescu, Existence of solutions for nonconvex functional differential inclusions, E. J. Diff. Equ., (2004), 6 pp.   Google Scholar

[27]

D. O'Regan, Integral inclusions of upper semi-continuous or lower semi-continuous type, Proc. Am. Math. Soc., 124 (1996), 2391-2399.  doi: 10.1090/S0002-9939-96-03456-9.  Google Scholar

[28]

B. E. Paden and S. S. Sastry, A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulator, IEEE Trans. Circuits Syst., 34 (1987), 73-82.  doi: 10.1109/TCS.1987.1086038.  Google Scholar

[29]

N. S. Papageorgiou, Periodic solutions of nonconvex differential inclusions, Appl. Math. Lett., 6 (1993), 99-101.  doi: 10.1016/0893-9659(93)90110-9.  Google Scholar

[30]

S. T. Qin and X. P. Xue, Periodic solutions for nonlinear differential inclusions with multivalued perturbations, J. Math. Anal. Appl., 424 (2015), 988-1005.  doi: 10.1016/j.jmaa.2014.11.057.  Google Scholar

[31]

A. V. Surkov, On the stability of functional-differential inclusions with the use of invariantly differentiable Lyapunov functionals, Differential Equations, 43 (2007), 1079-1087.  doi: 10.1134/S001226610708006X.  Google Scholar

[32]

D. Turkoglu and I. Altun, A fixed point theorem for multi-valued mappings and its applications to integral inclusions, Appl. Math. Lett., 20 (2007), 563-570.  doi: 10.1016/j.aml.2006.07.002.  Google Scholar

[33]

K. N. Wang and A. N. Michel, Stability analysis of differential inclusions in banach space with application to nonlinear systems with time delays, IEEE Trans. Circuits Syst. I, 43 (1996), 617-626.  doi: 10.1109/81.526677.  Google Scholar

[34]

X. P. Xue and J. F. Yu, Periodic solutions for semi-linear evolution inclusions, J. Math. Anal. Appl., 331 (2007), 1246-1262.  doi: 10.1016/j.jmaa.2006.09.056.  Google Scholar

[35]

P. Zecca and P. L. Zezza, Nonlinear boundary value problems in Banach spaces for multivalued differential equations in noncompact intervals, Nonlinear Anal., 3 (1979), 347-352.  doi: 10.1016/0362-546X(79)90024-5.  Google Scholar

show all references

References:
[1]

J.-P. Aubin and A. Cellina, Differential Inclusions, Set-Valued Functions and Viability Theory, Grundlehren der Mathematischen Wissenschaften, 264. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[3]

M. Benchohra and S. K. Ntouyas, Existence results for functional differential inclusions, E. J. Diff. Equ., (2001), 8 pp.   Google Scholar

[4]

A. Boucherif and C. C. Tisdell, Existence of periodic and non-periodic solutions to systems of boundary value problems for first-order differential inclusions with super-linear growth, Appl. Math. Comput., 204 (2008), 441-449.  doi: 10.1016/j.amc.2008.07.001.  Google Scholar

[5]

Z. W. Cai, J. H. Huang and L. H. Huang, Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 3591-3614.  doi: 10.3934/dcdsb.2017181.  Google Scholar

[6]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[7]

J. Cortés, Discontinuous dynamical systems: A tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Syst. Mag., 28 (2008), 36-73.  doi: 10.1109/MCS.2008.919306.  Google Scholar

[8]

F. S. De Blasi, L. Górniewicz and G. Pianigiani, Topological degree and periodic solutions of differential inclusions, Nonlinear Anal., 37 (1999), 217-245.  doi: 10.1016/S0362-546X(98)00044-3.  Google Scholar

[9]

B. C. Dhage, Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications, Comput. Math. Appl., 51 (2006), 589-604.  doi: 10.1016/j.camwa.2005.07.017.  Google Scholar

[10]

J. Dugundji and A. Granas, Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.  Google Scholar

[11]

A. F. Filippov, Differential Equations with Discontinuous Right-hand Side, Mathematics and Its Applications (Soviet Series), 18. Kluwer Academic, Boston, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[12]

M. Forti, M. Grazzini, P. Nistri and L. Pancioni, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Phys. D, 214 (2006), 88-99.  doi: 10.1016/j.physd.2005.12.006.  Google Scholar

[13]

G. Haddad, Monotone viable trajectories for functional differential inclusions, J. Differential Equations, 42 (1981), 1-24.  doi: 10.1016/0022-0396(81)90031-0.  Google Scholar

[14]

G. Haddad, Topological properties of the sets of solutions for functional differential inclusions, Nonlinear Anal., 5 (1981), 1349-1366.  doi: 10.1016/0362-546X(81)90111-5.  Google Scholar

[15]

J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[16]

H. Hermes, Discontinuous Vector Fields And Feedback Control, Differential Equations and Dynamical Systems, Academic, New York, (1967), 155–165.  Google Scholar

[17]

S. H. Hong, Existence results for functional differential inclusions with infinite delay, Acta Math. Sin., 22 (2006), 773-780.  doi: 10.1007/s10114-005-0600-y.  Google Scholar

[18]

S. H. Hong and L. Wang, Existence of solutions for integral inclusions, J. Math. Anal. Appl., 317 (2006), 429-441.  doi: 10.1016/j.jmaa.2006.01.057.  Google Scholar

[19]

S. C. Hu and N. S. Papageorgiou, On the existence of periodic solutions for nonconvex valued differential inclusions in $R^{N}$, Proc. Amer. Math. Soc., 123 (1995), 3043-3050.  doi: 10.2307/2160658.  Google Scholar

[20]

S. C. Hu, D. A. Kandilakis and N. S. Papageorgiou, Periodic solutions for nonconvex differential inclusions, Proc. Amer. Math. Soc., 127 (1999), 89-94.  doi: 10.1090/S0002-9939-99-04338-5.  Google Scholar

[21]

J. P. Lasalle, The Stability of Dynamical System, SIAM, Philadelphia, 1976.  Google Scholar

[22]

A. Lasota and Z. Opial, An application of the Kukutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys., 13 (1965), 781-786.   Google Scholar

[23]

Y. Li and Z. Lin, Periodic solutions of differential inclusions, Nonlinear Anal., 24 (1995), 631-641.  doi: 10.1016/0362-546X(94)00111-T.  Google Scholar

[24]

G. C. Li and X. P. Xue, On the existence of periodic solutions for differential inclusions, J. Math. Anal. Appl., 276 (2002), 168-183.  doi: 10.1016/S0022-247X(02)00397-9.  Google Scholar

[25]

K.-Z. Liu, X.-M. Sun, J. Liu and A. R. Teel, Stability theorems for delay differential inclusions, IEEE Trans. Autom. Control, 61 (2016), 3215-3220.  doi: 10.1109/TAC.2015.2507782.  Google Scholar

[26]

V. Lupulescu, Existence of solutions for nonconvex functional differential inclusions, E. J. Diff. Equ., (2004), 6 pp.   Google Scholar

[27]

D. O'Regan, Integral inclusions of upper semi-continuous or lower semi-continuous type, Proc. Am. Math. Soc., 124 (1996), 2391-2399.  doi: 10.1090/S0002-9939-96-03456-9.  Google Scholar

[28]

B. E. Paden and S. S. Sastry, A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulator, IEEE Trans. Circuits Syst., 34 (1987), 73-82.  doi: 10.1109/TCS.1987.1086038.  Google Scholar

[29]

N. S. Papageorgiou, Periodic solutions of nonconvex differential inclusions, Appl. Math. Lett., 6 (1993), 99-101.  doi: 10.1016/0893-9659(93)90110-9.  Google Scholar

[30]

S. T. Qin and X. P. Xue, Periodic solutions for nonlinear differential inclusions with multivalued perturbations, J. Math. Anal. Appl., 424 (2015), 988-1005.  doi: 10.1016/j.jmaa.2014.11.057.  Google Scholar

[31]

A. V. Surkov, On the stability of functional-differential inclusions with the use of invariantly differentiable Lyapunov functionals, Differential Equations, 43 (2007), 1079-1087.  doi: 10.1134/S001226610708006X.  Google Scholar

[32]

D. Turkoglu and I. Altun, A fixed point theorem for multi-valued mappings and its applications to integral inclusions, Appl. Math. Lett., 20 (2007), 563-570.  doi: 10.1016/j.aml.2006.07.002.  Google Scholar

[33]

K. N. Wang and A. N. Michel, Stability analysis of differential inclusions in banach space with application to nonlinear systems with time delays, IEEE Trans. Circuits Syst. I, 43 (1996), 617-626.  doi: 10.1109/81.526677.  Google Scholar

[34]

X. P. Xue and J. F. Yu, Periodic solutions for semi-linear evolution inclusions, J. Math. Anal. Appl., 331 (2007), 1246-1262.  doi: 10.1016/j.jmaa.2006.09.056.  Google Scholar

[35]

P. Zecca and P. L. Zezza, Nonlinear boundary value problems in Banach spaces for multivalued differential equations in noncompact intervals, Nonlinear Anal., 3 (1979), 347-352.  doi: 10.1016/0362-546X(79)90024-5.  Google Scholar

Figure 1.  Discontinuous activation functions of Example 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Time-domain behaviors of the state variables $ x_{1}(t) $ and $ x_{2}(t) $ of system (75) in Example 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Time evolution of variables $ x_{1}(t) $ for system (75) under the switching state-feedback controller (48) in Example 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Time evolution of variables $ x_{2}(t) $ for system (75) under the switching state-feedback controller (48) in Example 1
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021
[2]

Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251
[3]

Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial & Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061
[4]

Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158
[5]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263
[6]

Richard H. Rand, Asok K. Sen. A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Communications on Pure & Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567
[7]

Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks & Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297
[8]

Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779
[9]

Linna Li, Changjun Yu, Ning Zhang, Yanqin Bai, Zhiyuan Gao. A time-scaling technique for time-delay switched systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1825-1843. doi: 10.3934/dcdss.2020108
[10]

B. Cantó, C. Coll, A. Herrero, E. Sánchez, N. Thome. Pole-assignment of discrete time-delay systems with symmetries. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 641-649. doi: 10.3934/dcdsb.2006.6.641
[11]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113
[12]

Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021076
[13]

Ming He, Xiaoyun Ma, Weijiang Zhang. Oscillation death in systems of oscillators with transferable coupling and time-delay. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 737-745. doi: 10.3934/dcds.2001.7.737
[14]

Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial & Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471
[15]

Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, 2021, 29 (5) : 2973-2985. doi: 10.3934/era.2021022
[16]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[17]

Jing Li, Gui-Quan Sun, Zhen Jin. Interactions of time delay and spatial diffusion induce the periodic oscillation of the vegetation system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021127
[18]

Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329
[19]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072
[20]

J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (313)
  • HTML views (162)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy