## Stabilization strategies of supply networks with stochastic switched topology.(English)Zbl 1271.93121

Summary: In this paper, a dynamical supply networks model with stochastic switched topology is presented, in which the stochastic switched topology is dependent on a continuous time Markov process. The goal is to design the state-feedback control strategies to stabilize the dynamical supply networks. Based on Lyapunov stability theory, sufficient conditions for the existence of state feedback control strategies are given in terms of matrix inequalities, which ensure the robust stability of the supply networks at the stationary states and a prescribed $$H_\infty$$ disturbance attenuation level with respect to the uncertain demand. A numerical example is given to illustrate the effectiveness of the proposed method.

### MSC:

 93D15 Stabilization of systems by feedback 90B15 Stochastic network models in operations research 93B36 $$H^\infty$$-control
Full Text:

### References:

 [1] C. F. Daganzo, A Theory of Supply Chains, vol. 526, Springer, Berlin, Germany, 2003. · Zbl 1030.90003 [2] T. Nagatani and D. Helbing, “Stability analysis and stabilization strategies for linear supply chains,” Physica A, vol. 335, no. 3, pp. 644-660, 2004. [3] A. Surana, S. Kumara, M. Greaves, and U. N. Raghavan, “Supply-chain networks: a complex adaptive systems perspective,” International Journal of Production Research, vol. 43, no. 20, pp. 4235-4265, 2005. [4] D. Helbing, D. Armbruster, A. S. Mikhailov, and E. Lefeber, “Information and material flows in complex networks,” Physica A, vol. 363, no. 1, pp. 11-16, 2006. [5] D. Helbing, S. Lämmer, T. Seidel, P. \vSeba, and T. Płatkowski, “Physics, stability, and dynamics of supply networks,” Physical Review E, vol. 70, no. 6, Article ID 066116, 6 pages, 2004. [6] C. F. Daganzo, “On the stability of supply chains,” Operations Research, vol. 52, no. 6, pp. 909-921, 2004. · Zbl 1165.90346 [7] M. Laumanns and E. Lefeber, “Robust optimal control of material flows in demand-driven supply networks,” Physica A, vol. 363, no. 1, pp. 24-31, 2006. [8] Y. F. Ouyang and X. P. Li, “The bullwhip effect in supply chain networks,” European Journal of Operational Research, vol. 201, no. 3, pp. 799-810, 2010. · Zbl 1173.90338 [9] G. F. Yang, Z. P. Wang, and X. Q. Li, “The optimization of the closed-loop supply chain network,” Transportation Research E, vol. 45, no. 1, pp. 16-28, 2009. [10] M. Dong, F. L. He, and X. F. Shao, “Modeling and analysis of material flows in re-entrant supply chain networks using modified partial differential equations,” Journal of Applied Mathematics, vol. 2011, Article ID 325690, 14 pages, 2011. · Zbl 1216.35155 [11] L. Zhang and Y. Zhou, “A new approach to supply chain network equilibrium models,” Computers & Industrial Engineering, vol. 63, no. 1, pp. 82-88, 2012. [12] Y. R. Liu, Z. D. Wang, and X. H. Liu, “Exponential synchronization of complex networks with Markovian jump and mixed delays,” Physics Letters A, vol. 372, no. 22, pp. 3986-3998, 2008. · Zbl 1220.90040 [13] H. J. Li and D. Yue, “Synchronization of Markovian jumping stochastic complex networks with distributed time delays and probabilistic interval discrete time-varying delays,” Journal of Physics A, vol. 43, no. 10, Article ID 105101, 2010. · Zbl 1198.60040 [14] Y. Ouyang and C. Daganzo, “Robust tests for the bullwhip effect in supply chains with stochastic dynamics,” European Journal of Operational Research, vol. 185, no. 1, pp. 340-353, 2008. · Zbl 1137.90334 [15] Y. Wu, M. Dong, W. Tang, and F. F. Chen, “Performance analysis of serial supply chain networks considering system disruptions,” Production Planning and Control, vol. 21, no. 8, pp. 774-793, 2010. [16] L. Rodrigues and E. K. Boukas, “Piecewise-linear H\infty controller synthesis with applications to inventory control of switched production systems,” Automatica, vol. 42, no. 8, pp. 1245-1254, 2006. · Zbl 1097.90006 [17] S. Li, W. Tang, and J. Zhang, “Robust control for synchronization of singular complex delayed networks with stochastic switched coupling,” International Journal of Computer Mathematics, vol. 89, no. 10, pp. 1332-1344, 2012. · Zbl 1255.93105 [18] E. Adida and G. Perakis, “A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders,” Naval Research Logistics, vol. 54, no. 7, pp. 767-795, 2007. · Zbl 1137.90302 [19] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, SIAM, Philadelphia, Pa, USA, 1994. · Zbl 0816.93004 [20] J. Zhang and W. Tang, “Output feedback optimal guaranteed cost control of uncertain piecewise linear systems,” International Journal of Robust and Nonlinear Control, vol. 19, no. 5, pp. 569-590, 2009. · Zbl 1160.93342 [21] J. X. Zhang and W. S. Tang, “Optimal control for a class of chaotic systems,” Journal of Applied Mathematics, vol. 2012, Article ID 859542, 20 pages, 2012. · Zbl 1245.49044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.