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November  2017, 22(9): 3591-3614. doi: 10.3934/dcdsb.2017181

Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks

Zuowei Cai a,b, , Jianhua Huang a,, and  Lihong Huang c,

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

* Corresponding author: Jianhua Huang

Received  September 2016 Revised  June 2017 Published  July 2017

Fund Project: The first author is supported by NSF of China(No.11626100), Natural Science Foundation of Hunan Province(No.2016JJ3078) and Scientific Research Youth Project of Hunan Provincial Education Department(No.16B133).

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.

Keywords: Retarded differential inclusions(RDI), Filippov solution, stability, Razumikhin techniques, almost indefinite Lyapunov-like function.
Mathematics Subject Classification: Primary:34D20, 34K09;Secondary:34K20.
Citation: Zuowei Cai, Jianhua Huang, Lihong Huang. Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3591-3614. doi: 10.3934/dcdsb.2017181
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.

Figure 1.  Discontinuous neuron activation functions of Example 1
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Figure 2.  Time-domain behaviors of the state variables $x_{1}(t)$ and $x_{2}(t)$ for system (50) under the switching state-feedback controller (53) in Example 1
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Figure 3.  Discontinuous neuron activation functions of Example 2
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Figure 4.  Time-domain behaviors of the state variables $x_{1}(t)$ and $x_{2}(t)$ for system (50) without external input in Example 2
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