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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 |
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.
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. |
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. |




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