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

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

* 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.

Citation: Zuowei Cai, Jianhua Huang, Lihong Huang. Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks. Discrete & 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.  Google Scholar

[2]

J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[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.  Google Scholar

[4]

M. Benchohra and S. K. Ntouyas, Existence results for functional differential inclusions, Electronic Journal of Differential Equations, 2001 (2001), 1-8.   Google Scholar

[5]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008.  Google Scholar

[6]

V. I. Blagodat-skik and A. F. Filippov, Differential inclusions and optimal control, Proc. Steklov Inst. Math., 4 (1986), 199-259.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Z. W. CaiL. H. HuangZ. 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.  Google Scholar

[10]

E. K. P. ChongS. 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.  Google Scholar

[11]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.  Google Scholar

[12]

F. H. Clarke, Y. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.  Google Scholar

[13]

J. Cortés, Discontinuous dynamical systems, IEEE Control Syst. Mag., 28 (2008), 36-73.  doi: 10.1109/MCS.2008.919306.  Google Scholar

[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.  Google Scholar

[15]

M. FortiM. GrazziniP. 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.  Google Scholar

[16]

M. FortiP. 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.  Google Scholar

[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.  Google Scholar

[18]

Z. Y. GuoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[22]

H. Hermes, Discontinuous vector fields and feedback control, in: Differential Equations and Dynamical Systems, Academic, New York, (1967), 155-165.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[27]

K. Z. LiuX. M. SunJ. 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

[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.  Google Scholar

[29]

X. Y. LiuJ. D. CaoW. 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.  Google Scholar

[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.  Google Scholar

[31]

V. Lupulescu, Existence of solutions for nonconvex functional differential inclusions, Electronic Journal of Differential Equations, 2004 (2004), 1-6.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

V. I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automat. Contr., 22 (1977), 212-222.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

S. P. WenT. W. HuangZ. G. ZengY. 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.  Google Scholar

[39]

A. L. WuS. 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.  Google Scholar

[40]

C. J. XuP. 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.  Google Scholar

[41]

X. S. YangD. W. C. HoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, MA: Birkhauser, Boston, 1990.  Google Scholar

[2]

J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[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.  Google Scholar

[4]

M. Benchohra and S. K. Ntouyas, Existence results for functional differential inclusions, Electronic Journal of Differential Equations, 2001 (2001), 1-8.   Google Scholar

[5]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008.  Google Scholar

[6]

V. I. Blagodat-skik and A. F. Filippov, Differential inclusions and optimal control, Proc. Steklov Inst. Math., 4 (1986), 199-259.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Z. W. CaiL. H. HuangZ. 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.  Google Scholar

[10]

E. K. P. ChongS. 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.  Google Scholar

[11]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.  Google Scholar

[12]

F. H. Clarke, Y. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.  Google Scholar

[13]

J. Cortés, Discontinuous dynamical systems, IEEE Control Syst. Mag., 28 (2008), 36-73.  doi: 10.1109/MCS.2008.919306.  Google Scholar

[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.  Google Scholar

[15]

M. FortiM. GrazziniP. 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.  Google Scholar

[16]

M. FortiP. 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.  Google Scholar

[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.  Google Scholar

[18]

Z. Y. GuoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[22]

H. Hermes, Discontinuous vector fields and feedback control, in: Differential Equations and Dynamical Systems, Academic, New York, (1967), 155-165.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[27]

K. Z. LiuX. M. SunJ. 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

[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.  Google Scholar

[29]

X. Y. LiuJ. D. CaoW. 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.  Google Scholar

[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.  Google Scholar

[31]

V. Lupulescu, Existence of solutions for nonconvex functional differential inclusions, Electronic Journal of Differential Equations, 2004 (2004), 1-6.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

V. I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automat. Contr., 22 (1977), 212-222.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

S. P. WenT. W. HuangZ. G. ZengY. 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.  Google Scholar

[39]

A. L. WuS. 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.  Google Scholar

[40]

C. J. XuP. 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.  Google Scholar

[41]

X. S. YangD. W. C. HoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Discontinuous neuron activation functions of Example 1
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
Figure 3.  Discontinuous neuron activation functions of Example 2
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
[1]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[2]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[3]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[4]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[5]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[6]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[7]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[8]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119

[9]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[10]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[11]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[12]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[13]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[14]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[15]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[16]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[17]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[18]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[19]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[20]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (197)
  • HTML views (68)
  • Cited by (14)

Other articles
by authors

[Back to Top]