# American Institute of Mathematical Sciences

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

## Periodicity and stabilization control of the delayed Filippov system with perturbation

 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.

Citation: Zuowei Cai, Jianhua Huang, Liu Yang, Lihong Huang. Periodicity and stabilization control of the delayed Filippov system with perturbation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1439-1467. doi: 10.3934/dcdsb.2019235
##### References:
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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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977. [16] H. Hermes, Discontinuous Vector Fields And Feedback Control, Differential Equations and Dynamical Systems, Academic, New York, (1967), 155–165. [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. [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. [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. [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. [21] J. P. Lasalle, The Stability of Dynamical System, SIAM, Philadelphia, 1976. [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. [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. [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. [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. [26] V. Lupulescu, Existence of solutions for nonconvex functional differential inclusions, E. J. Diff. Equ., (2004), 6 pp. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. [3] M. Benchohra and S. K. Ntouyas, Existence results for functional differential inclusions, E. J. Diff. Equ., (2001), 8 pp. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977. [16] H. Hermes, Discontinuous Vector Fields And Feedback Control, Differential Equations and Dynamical Systems, Academic, New York, (1967), 155–165. [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. [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. [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. [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. [21] J. P. Lasalle, The Stability of Dynamical System, SIAM, Philadelphia, 1976. [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. [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. [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. [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. [26] V. Lupulescu, Existence of solutions for nonconvex functional differential inclusions, E. J. Diff. Equ., (2004), 6 pp. [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. [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. [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. [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. [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. [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. [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. [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. [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.
Discontinuous activation functions of Example 1
Time-domain behaviors of the state variables $x_{1}(t)$ and $x_{2}(t)$ of system (75) in Example 1
Time evolution of variables $x_{1}(t)$ for system (75) under the switching state-feedback controller (48) in Example 1
Time evolution of variables $x_{2}(t)$ for system (75) under the switching state-feedback controller (48) in Example 1
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