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Expansivity implies existence of Hölder continuous Lyapunov function
Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks
a. | College of Science, National University of Defense Technology, Changsha, Hunan 410073, China |
b. | Department of Information Technology, Hunan Women's University, Changsha, Hunan 410002, China |
c. | School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114 China |
In this paper, a general class of nonlinear dynamical systems described by retarded differential equations with discontinuous right-hand sides is considered. Under the extended Filippov-framework, we investigate some basic stability problems for retarded differential inclusions (RDI) with given initial conditions by using the generalized Lyapunov-Razumikhin method. Comparing with the previous work, the main results given in this paper show that the Lyapunov-Razumikhin function is allowed to have a indefinite or positive definite derivative for almost everywhere along the solution trajectories of RDI. However, in most of the existing literature, the derivative (if it exists) of Lyapunov-Razumikhin function is required to be negative or semi-negative definite for almost everywhere. In addition, the Lyapunov-Razumikhin function in this paper is allowed to be non-smooth. To deal with the stability, we also drop the specific condition that the Lyapunov-Razumikhin function should have an infinitesimal upper limit. Finally, we apply the proposed Razumikhin techniques to handle the stability or stabilization control of discontinuous time-delayed neural networks. Meanwhile, we present two examples to demonstrate the effectiveness of the developed method.
References:
[1] |
J. P. Aubin and H. Frankowska,
Set-Valued Analysis, MA: Birkhauser, Boston, 1990. |
[2] |
J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-69512-4. |
[3] |
A. Baciotti and F. Ceragioli,
Stability and stabilization of discontinuous systems and non-smooth Lyapunov function, ESAIM Control Optim. Calc. Var., 4 (1999), 361-376.
doi: 10.1051/cocv:1999113. |
[4] |
M. Benchohra and S. K. Ntouyas,
Existence results for functional differential inclusions, Electronic Journal of Differential Equations, 2001 (2001), 1-8.
|
[5] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk,
Piecewise-smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. |
[6] |
V. I. Blagodat-skik and A. F. Filippov,
Differential inclusions and optimal control, Proc. Steklov Inst. Math., 4 (1986), 199-259.
|
[7] |
Z. W. Cai and L. H. Huang,
Finite-time stabilization of delayed memristive neural networks: Discontinuous state-feedback and adaptive control approach, IEEE Trans. Neural Netw. Learn. Syst., PP (2017), 1-13.
doi: 10.1109/TNNLS.2017.2651023. |
[8] |
Z. W. Cai and L. H. Huang,
Novel adaptive control and state-feedback control strategies to finite-time stabilization of discontinuous delayed networks, IEEE Trans. Syst., Man, Cybern., Syst., PP (2017), 1-11.
doi: 10.1109/TSMC.2017.2657784. |
[9] |
Z. W. Cai, L. H. Huang, Z. Y. Guo and X. Y. Chen,
On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions, Neural Networks, 33 (2012), 97-113.
doi: 10.1016/j.neunet.2012.04.009. |
[10] |
E. K. P. Chong, S. Hui and S. H. Zak,
An analysis of a class of neural networks for solving linear programming problems, IEEE Trans. Autom. Control, 44 (1999), 1995-2006.
doi: 10.1109/9.802909. |
[11] |
F. H. Clarke,
Optimization and Nonsmooth Analysis, Wiley, New York, 1983. |
[12] |
F. H. Clarke, Y. Ledyaev, R. J. Stern and P. R. Wolenski,
Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998. |
[13] |
J. Cortés,
Discontinuous dynamical systems, IEEE Control Syst. Mag., 28 (2008), 36-73.
doi: 10.1109/MCS.2008.919306. |
[14] |
A. F. Filippov,
Differential Equations with Discontinuous Right-hand Side, in: Mathematics and Its Applications (Soviet Series), Kluwer Academic, Boston, 1988.
doi: 10.1007/978-94-015-7793-9. |
[15] |
M. Forti, M. Grazzini, P. Nistri and L. Pancioni,
Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Physica D, 214 (2006), 88-99.
doi: 10.1016/j.physd.2005.12.006. |
[16] |
M. Forti, P. Nistri and M. Quincampoix,
Generalized neural network for nonsmooth nonlinear programming problems, IEEE Trans. Circuits Syst. I, 51 (2004), 1741-1754.
doi: 10.1109/TCSI.2004.834493. |
[17] |
Z. Y. Guo and L. H. Huang,
Generalized Lyapunov method for discontinuous systems, Nonlinear Anal., 71 (2009), 3083-3092.
doi: 10.1016/j.na.2009.01.220. |
[18] |
Z. Y. Guo, J. Wang and Z. Yan,
Attractivity analysis of memristor-based cellular neural networks with time-varying delays, IEEE Trans. Neural Netw. Learn. Syst., 25 (2014), 704-717.
doi: 10.1109/TNNLS.2013.2280556. |
[19] |
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. |
[20] |
G. Haddad,
Monotone viable trajectories for functional differential inclusions, J. Differential Equations, 42 (1981), 1-24.
doi: 10.1016/0022-0396(81)90031-0. |
[21] |
J. K. Hale,
Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. |
[22] |
H. Hermes, Discontinuous vector fields and feedback control, in: Differential Equations and Dynamical Systems, Academic, New York, (1967), 155-165. |
[23] |
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. |
[24] |
L. H. Huang, Z. Y. Guo and J. F. Wang, Theory and Applications of Differential Equations with Discontinuous Right-hand Sides (in Chinese), Science Press, Beijing, 2011. |
[25] |
M. P. Kennedy and L. O. Chua,
Neural networks for nonlinear programming, IEEE Trans. Circuits Syst. I, 35 (1988), 554-562.
doi: 10.1109/31.1783. |
[26] |
N. N. Krasovskii,
Stability of Motion. Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, CA: Stanford Univ. Press, Stanford, 1963. (Transl. from Russian by J. L. Brenner). |
[27] |
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. |
[28] |
K. Z. Liu and X. M. Sun,
Razumikhin-type theorems for hybrid system with memory, Automatica, 71 (2016), 72-77.
doi: 10.1016/j.automatica.2016.04.038. |
[29] |
X. Y. Liu, J. D. Cao, W. W. Yu and Q. Song,
Nonsmooth finite-time synchronization of switched coupled neural networks, IEEE Trans. Cybern., 46 (2016), 2360-2371.
doi: 10.1109/TCYB.2015.2477366. |
[30] |
A. C. J. Luo,
Discontinuous Dynamical Systems on Time-varying Domains, Higher Education Press, Beijing, 2009.
doi: 10.1007/978-3-642-00253-3. |
[31] |
V. Lupulescu,
Existence of solutions for nonconvex functional differential inclusions, Electronic Journal of Differential Equations, 2004 (2004), 1-6.
|
[32] |
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. |
[33] |
T. T. Su and X. S. Yang,
Finite-time synchronization of competitive neural networks with mixed delays, Discrete & Continuous Dynamical Systems-Series B, 21 (2016), 3655-3667.
doi: 10.3934/dcdsb.2016115. |
[34] |
A. 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. |
[35] |
V. I. Utkin,
Variable structure systems with sliding modes, IEEE Trans. Automat. Contr., 22 (1977), 212-222.
|
[36] |
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. |
[37] |
L. Wang and Y. Shen,
Finite-time stabilizability and instabilizability of delayed memristive neural networks with nonlinear discontinuous controller, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 2914-2924.
doi: 10.1109/TNNLS.2015.2460239. |
[38] |
S. P. Wen, T. W. Huang, Z. G. Zeng, Y. R. Chen and P. Li,
Circuit design and exponential stabilization of memristive neural networks, Neural Networks, 63 (2015), 48-56.
doi: 10.1016/j.neunet.2014.10.011. |
[39] |
A. L. Wu, S. P. Wen and Z. G. Zeng,
Synchronization control of a class of memristor-based recurrent neural networks, Information Sciences, 183 (2012), 106-116.
doi: 10.1016/j.ins.2011.07.044. |
[40] |
C. J. Xu, P. L. Li and Y. C. Pang,
Exponential stability of almost periodic solutions for memristor-based neural networks with distributed leakage delays, Neural Comput., 28 (2016), 2726-2756.
doi: 10.1162/NECO_a_00895. |
[41] |
X. S. Yang, D. W. C. Ho, J. Q. Lu and Q. Song,
Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays, IEEE Trans. Fuzzy Syst., 23 (2015), 2302-2316.
doi: 10.1109/TFUZZ.2015.2417973. |
[42] |
X. S. Yang and D. W. C. Ho,
Synchronization of delayed memristive neural networks: Robust analysis approach, IEEE Trans. Cybern., 46 (2016), 3377-3387.
doi: 10.1109/TCYB.2015.2505903. |
[43] |
B. Zhou and A. V. Egorov,
Razumikhin and Krasovskii stability theorems for time-varying time-delay systems, Automatica, 71 (2016), 281-291.
doi: 10.1016/j.automatica.2016.04.048. |
show all references
References:
[1] |
J. P. Aubin and H. Frankowska,
Set-Valued Analysis, MA: Birkhauser, Boston, 1990. |
[2] |
J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-69512-4. |
[3] |
A. Baciotti and F. Ceragioli,
Stability and stabilization of discontinuous systems and non-smooth Lyapunov function, ESAIM Control Optim. Calc. Var., 4 (1999), 361-376.
doi: 10.1051/cocv:1999113. |
[4] |
M. Benchohra and S. K. Ntouyas,
Existence results for functional differential inclusions, Electronic Journal of Differential Equations, 2001 (2001), 1-8.
|
[5] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk,
Piecewise-smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. |
[6] |
V. I. Blagodat-skik and A. F. Filippov,
Differential inclusions and optimal control, Proc. Steklov Inst. Math., 4 (1986), 199-259.
|
[7] |
Z. W. Cai and L. H. Huang,
Finite-time stabilization of delayed memristive neural networks: Discontinuous state-feedback and adaptive control approach, IEEE Trans. Neural Netw. Learn. Syst., PP (2017), 1-13.
doi: 10.1109/TNNLS.2017.2651023. |
[8] |
Z. W. Cai and L. H. Huang,
Novel adaptive control and state-feedback control strategies to finite-time stabilization of discontinuous delayed networks, IEEE Trans. Syst., Man, Cybern., Syst., PP (2017), 1-11.
doi: 10.1109/TSMC.2017.2657784. |
[9] |
Z. W. Cai, L. H. Huang, Z. Y. Guo and X. Y. Chen,
On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions, Neural Networks, 33 (2012), 97-113.
doi: 10.1016/j.neunet.2012.04.009. |
[10] |
E. K. P. Chong, S. Hui and S. H. Zak,
An analysis of a class of neural networks for solving linear programming problems, IEEE Trans. Autom. Control, 44 (1999), 1995-2006.
doi: 10.1109/9.802909. |
[11] |
F. H. Clarke,
Optimization and Nonsmooth Analysis, Wiley, New York, 1983. |
[12] |
F. H. Clarke, Y. Ledyaev, R. J. Stern and P. R. Wolenski,
Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998. |
[13] |
J. Cortés,
Discontinuous dynamical systems, IEEE Control Syst. Mag., 28 (2008), 36-73.
doi: 10.1109/MCS.2008.919306. |
[14] |
A. F. Filippov,
Differential Equations with Discontinuous Right-hand Side, in: Mathematics and Its Applications (Soviet Series), Kluwer Academic, Boston, 1988.
doi: 10.1007/978-94-015-7793-9. |
[15] |
M. Forti, M. Grazzini, P. Nistri and L. Pancioni,
Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Physica D, 214 (2006), 88-99.
doi: 10.1016/j.physd.2005.12.006. |
[16] |
M. Forti, P. Nistri and M. Quincampoix,
Generalized neural network for nonsmooth nonlinear programming problems, IEEE Trans. Circuits Syst. I, 51 (2004), 1741-1754.
doi: 10.1109/TCSI.2004.834493. |
[17] |
Z. Y. Guo and L. H. Huang,
Generalized Lyapunov method for discontinuous systems, Nonlinear Anal., 71 (2009), 3083-3092.
doi: 10.1016/j.na.2009.01.220. |
[18] |
Z. Y. Guo, J. Wang and Z. Yan,
Attractivity analysis of memristor-based cellular neural networks with time-varying delays, IEEE Trans. Neural Netw. Learn. Syst., 25 (2014), 704-717.
doi: 10.1109/TNNLS.2013.2280556. |
[19] |
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. |
[20] |
G. Haddad,
Monotone viable trajectories for functional differential inclusions, J. Differential Equations, 42 (1981), 1-24.
doi: 10.1016/0022-0396(81)90031-0. |
[21] |
J. K. Hale,
Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. |
[22] |
H. Hermes, Discontinuous vector fields and feedback control, in: Differential Equations and Dynamical Systems, Academic, New York, (1967), 155-165. |
[23] |
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. |
[24] |
L. H. Huang, Z. Y. Guo and J. F. Wang, Theory and Applications of Differential Equations with Discontinuous Right-hand Sides (in Chinese), Science Press, Beijing, 2011. |
[25] |
M. P. Kennedy and L. O. Chua,
Neural networks for nonlinear programming, IEEE Trans. Circuits Syst. I, 35 (1988), 554-562.
doi: 10.1109/31.1783. |
[26] |
N. N. Krasovskii,
Stability of Motion. Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, CA: Stanford Univ. Press, Stanford, 1963. (Transl. from Russian by J. L. Brenner). |
[27] |
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. |
[28] |
K. Z. Liu and X. M. Sun,
Razumikhin-type theorems for hybrid system with memory, Automatica, 71 (2016), 72-77.
doi: 10.1016/j.automatica.2016.04.038. |
[29] |
X. Y. Liu, J. D. Cao, W. W. Yu and Q. Song,
Nonsmooth finite-time synchronization of switched coupled neural networks, IEEE Trans. Cybern., 46 (2016), 2360-2371.
doi: 10.1109/TCYB.2015.2477366. |
[30] |
A. C. J. Luo,
Discontinuous Dynamical Systems on Time-varying Domains, Higher Education Press, Beijing, 2009.
doi: 10.1007/978-3-642-00253-3. |
[31] |
V. Lupulescu,
Existence of solutions for nonconvex functional differential inclusions, Electronic Journal of Differential Equations, 2004 (2004), 1-6.
|
[32] |
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. |
[33] |
T. T. Su and X. S. Yang,
Finite-time synchronization of competitive neural networks with mixed delays, Discrete & Continuous Dynamical Systems-Series B, 21 (2016), 3655-3667.
doi: 10.3934/dcdsb.2016115. |
[34] |
A. 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. |
[35] |
V. I. Utkin,
Variable structure systems with sliding modes, IEEE Trans. Automat. Contr., 22 (1977), 212-222.
|
[36] |
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. |
[37] |
L. Wang and Y. Shen,
Finite-time stabilizability and instabilizability of delayed memristive neural networks with nonlinear discontinuous controller, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 2914-2924.
doi: 10.1109/TNNLS.2015.2460239. |
[38] |
S. P. Wen, T. W. Huang, Z. G. Zeng, Y. R. Chen and P. Li,
Circuit design and exponential stabilization of memristive neural networks, Neural Networks, 63 (2015), 48-56.
doi: 10.1016/j.neunet.2014.10.011. |
[39] |
A. L. Wu, S. P. Wen and Z. G. Zeng,
Synchronization control of a class of memristor-based recurrent neural networks, Information Sciences, 183 (2012), 106-116.
doi: 10.1016/j.ins.2011.07.044. |
[40] |
C. J. Xu, P. L. Li and Y. C. Pang,
Exponential stability of almost periodic solutions for memristor-based neural networks with distributed leakage delays, Neural Comput., 28 (2016), 2726-2756.
doi: 10.1162/NECO_a_00895. |
[41] |
X. S. Yang, D. W. C. Ho, J. Q. Lu and Q. Song,
Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays, IEEE Trans. Fuzzy Syst., 23 (2015), 2302-2316.
doi: 10.1109/TFUZZ.2015.2417973. |
[42] |
X. S. Yang and D. W. C. Ho,
Synchronization of delayed memristive neural networks: Robust analysis approach, IEEE Trans. Cybern., 46 (2016), 3377-3387.
doi: 10.1109/TCYB.2015.2505903. |
[43] |
B. Zhou and A. V. Egorov,
Razumikhin and Krasovskii stability theorems for time-varying time-delay systems, Automatica, 71 (2016), 281-291.
doi: 10.1016/j.automatica.2016.04.048. |




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