# American Institute of Mathematical Sciences

September  2015, 5(3): 697-719. doi: 10.3934/mcrf.2015.5.697

## Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074 2 Department of Mathematics, Wayne State University, Detroit, Michigan 48202 3 Department of Electrical and Computer Engineering, Wayne State University, MI 48202, United States

Received  March 2014 Revised  June 2014 Published  July 2015

This work develops moment exponential stability of functional differential equations (FDEs) with Markovian switching, in which a two-time-scale (real time $t$ and fast time $t/\epsilon$ with a small parameter $\epsilon>0$) continuous-time and finite-state Markov chain is used to represent the switching process. The essence is that there is a time-scale separation, which is motivated by the consideration of networked control systems and manufacturing systems. Under suitable conditions, we establish a Razumikhin-type theorem on the $p$th moment exponential $\epsilon$-stability for the small parameter $\epsilon$. By virtue of the Razumikhin-type theorem, we further deduce mean-square exponential stability results for delay differential equations (DDEs) and ordinary differential equations (ODEs) with two-time-scale Markovian switching. These stability results show that the overall system may be stabilized by the Markov switching even when some of the underlying subsystems are unstable. It is noted that in the presence of the Markovian switching, the stationary distribution of the fast changing part of the Markov chain plays an important role. Explicit conditions for the mean-square exponential stability of linear equations are derived and illustrative examples are provided to demonstrate our results.
Citation: Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control and Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697
##### References:
 [1] M. Abundo and L. Caramellino, Some remarks on a Markov chain modelling cooperative biological systems, Open Systems & Information Dynamics, 3 (1995), 325-343. doi: 10.1007/BF02228996. [2] G. Badowski and G. Yin, Stability of hybrid dynamic systems containing singularly perturbed random processes, IEEE Transactions on Automatic Control, 47 (2002), 2021-2032. doi: 10.1109/TAC.2002.805682. [3] G. Barone-Adesi and R. Whaley, Efficient analytic approximation of American option values, Journal of Finance, 42 (1987), 301-320. doi: 10.1111/j.1540-6261.1987.tb02569.x. [4] M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation, PLoS Computational Biology, 2 (2006), 1017-1039. [5] A. Bensoussan, Z. Yan and G. Yin, Threshold-type policies for real options using regime-switching models, SIAM J. Financial Math., 3 (2012), 667-689. doi: 10.1137/110833300. [6] G. B. Di Masi, Y. M. Kabanov and W. J. Runggaldier, Mean variance hedging of options on stocks with Markov volatility, Theory of Probability and Applications, 39 (1994), 172-182. doi: 10.1137/1139008. [7] P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465. doi: 10.1016/j.insmatheco.2008.09.001. [8] J. R. Haddock, T. Krisztin, J. Terjéki and J. H. Wu., An invariance principle of Lyapunov-Razumikhin type for neutral functional differential equations, Journal of Differential Equations, 107 (1994), 395-417. doi: 10.1006/jdeq.1994.1019. [9] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [10] I. I. Kac and N. N. Krasovskii, On the stability of systems with random parameters, J. Appl. Math. Mech., 24 (1960), 1225-1246. [11] I. Karafyllis, P. Pepe and Z. P. Jiang, Input-to-output stability for systems described by retarded functional differential equations, European Journal of Control, 14 (2008), 539-555. doi: 10.3166/ejc.14.539-555. [12] R. Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen a/d Rijn, 1980. [13] R. Z. Khasminskii, C. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Processes and their Applications, 117 (2007), 1037-1051. doi: 10.1016/j.spa.2006.12.001. [14] R. Z. Khasminskii and G. Yin, On averaging principle: An asymptotic expansion approach, SIAM Journal on Mathematical Analysis, 35 (2004), 1534-1560. doi: 10.1137/S0036141002403973. [15] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986. [16] V. B. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Dordrecht, The Netherlands: Kluwer, 1992. doi: 10.1007/978-94-015-8084-7. [17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [18] H. J. Kushner, On the stability of processes defined by stochastic difference-differential equations, Journal of Differential Equations, 4 (1968), 424-443. doi: 10.1016/0022-0396(68)90028-4. [19] H. J. Kushner, Stability and existence of diffusions with discontinuous or rapidly growing drift terms, Journal of Differential Equations, 11 (1972), 156-168. doi: 10.1016/0022-0396(72)90086-1. [20] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984. [21] H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, MA, 1990. doi: 10.1007/978-1-4612-4482-0. [22] Q. Luo and X. Mao, Stochastic population dynamics under regime switching II, Journal of Mathematical Analysis and Applications, 355 (2009), 577-593. doi: 10.1016/j.jmaa.2009.02.010. [23] M. Jankovic, Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems, IEEE Transactions on Automatic Control, 46 (2001), 1048-1060. doi: 10.1109/9.935057. [24] X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. doi: 10.1533/9780857099402. [25] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Processes and their Applications, 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2. [26] X. Mao, Stochastic functional differential equations with Markovian switching, Functional Differential Equations, 6 (1999), 375-396. [27] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, London, Imperial College Press, 2006. doi: 10.1142/p473. [28] A. R. Teel, Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Transactions on Automatic Control, 43 (1998), 960-964. doi: 10.1109/9.701099. [29] F. Wu, G. Yin and L. Y. Wang, Moment exponential stability of random delay systems with two-time-scale Markovian switching, Nonlinear Analysis Series B: Real World Applications, 13 (2012), 2476-2490. doi: 10.1016/j.nonrwa.2012.02.013. [30] F. Wu, G. Yin and L. Y. Wang, Stability of a Pure Random Delay System with Two-Time-Scale Markovian Switching, Journal of Differential Equations, 253 (2012), 878-905. doi: 10.1016/j.jde.2012.04.017. [31] G. Yin, Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process, Asymptotic Analysis, 65 (2009), 203-222. [32] G. Yin, S. Kan, L. Y. Wang and C. Xu, Identification of systems with regime switching and unmodeled dynamics, IEEE Transaction on Automatical Control, 54 (2009), 34-47. doi: 10.1109/TAC.2008.2009487. [33] G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Singular Perturbation Approach, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0627-9. [34] G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382. doi: 10.1137/110851171. [35] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [36] C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Processes and their Applications, 103 (2003), 277-291. doi: 10.1016/S0304-4149(02)00230-2. [37] C. Yuan and G. Yin, Stability of hybrid stochastic delay systems whose discrete components have a large state space: A two-time- scale approach, Journal of Mathematical Analysis and Applications, 368 (2010), 103-119. doi: 10.1016/j.jmaa.2010.02.053. [38] Q. Zhang, Hybrid filtering for linear systems with non-Gaussian disturbances, IEEE Transaction on Automatical Control, 45 (2000), 50-61. doi: 10.1109/9.827355. [39] X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583. [40] C. Zhu, G. Yin and Q. S. Song, Stability of random- switching systems of differential equations, Quarterly of Applied Mathematics, 67 (2009), 201-220. doi: 10.1090/S0033-569X-09-01092-8.

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##### References:
 [1] M. Abundo and L. Caramellino, Some remarks on a Markov chain modelling cooperative biological systems, Open Systems & Information Dynamics, 3 (1995), 325-343. doi: 10.1007/BF02228996. [2] G. Badowski and G. Yin, Stability of hybrid dynamic systems containing singularly perturbed random processes, IEEE Transactions on Automatic Control, 47 (2002), 2021-2032. doi: 10.1109/TAC.2002.805682. [3] G. Barone-Adesi and R. Whaley, Efficient analytic approximation of American option values, Journal of Finance, 42 (1987), 301-320. doi: 10.1111/j.1540-6261.1987.tb02569.x. [4] M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation, PLoS Computational Biology, 2 (2006), 1017-1039. [5] A. Bensoussan, Z. Yan and G. Yin, Threshold-type policies for real options using regime-switching models, SIAM J. Financial Math., 3 (2012), 667-689. doi: 10.1137/110833300. [6] G. B. Di Masi, Y. M. Kabanov and W. J. Runggaldier, Mean variance hedging of options on stocks with Markov volatility, Theory of Probability and Applications, 39 (1994), 172-182. doi: 10.1137/1139008. [7] P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465. doi: 10.1016/j.insmatheco.2008.09.001. [8] J. R. Haddock, T. Krisztin, J. Terjéki and J. H. Wu., An invariance principle of Lyapunov-Razumikhin type for neutral functional differential equations, Journal of Differential Equations, 107 (1994), 395-417. doi: 10.1006/jdeq.1994.1019. [9] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [10] I. I. Kac and N. N. Krasovskii, On the stability of systems with random parameters, J. Appl. Math. Mech., 24 (1960), 1225-1246. [11] I. Karafyllis, P. Pepe and Z. P. Jiang, Input-to-output stability for systems described by retarded functional differential equations, European Journal of Control, 14 (2008), 539-555. doi: 10.3166/ejc.14.539-555. [12] R. Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen a/d Rijn, 1980. [13] R. Z. Khasminskii, C. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Processes and their Applications, 117 (2007), 1037-1051. doi: 10.1016/j.spa.2006.12.001. [14] R. Z. Khasminskii and G. Yin, On averaging principle: An asymptotic expansion approach, SIAM Journal on Mathematical Analysis, 35 (2004), 1534-1560. doi: 10.1137/S0036141002403973. [15] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986. [16] V. B. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Dordrecht, The Netherlands: Kluwer, 1992. doi: 10.1007/978-94-015-8084-7. [17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [18] H. J. Kushner, On the stability of processes defined by stochastic difference-differential equations, Journal of Differential Equations, 4 (1968), 424-443. doi: 10.1016/0022-0396(68)90028-4. [19] H. J. Kushner, Stability and existence of diffusions with discontinuous or rapidly growing drift terms, Journal of Differential Equations, 11 (1972), 156-168. doi: 10.1016/0022-0396(72)90086-1. [20] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984. [21] H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, MA, 1990. doi: 10.1007/978-1-4612-4482-0. [22] Q. Luo and X. Mao, Stochastic population dynamics under regime switching II, Journal of Mathematical Analysis and Applications, 355 (2009), 577-593. doi: 10.1016/j.jmaa.2009.02.010. [23] M. Jankovic, Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems, IEEE Transactions on Automatic Control, 46 (2001), 1048-1060. doi: 10.1109/9.935057. [24] X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. doi: 10.1533/9780857099402. [25] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Processes and their Applications, 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2. [26] X. Mao, Stochastic functional differential equations with Markovian switching, Functional Differential Equations, 6 (1999), 375-396. [27] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, London, Imperial College Press, 2006. doi: 10.1142/p473. [28] A. R. Teel, Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Transactions on Automatic Control, 43 (1998), 960-964. doi: 10.1109/9.701099. [29] F. Wu, G. Yin and L. Y. Wang, Moment exponential stability of random delay systems with two-time-scale Markovian switching, Nonlinear Analysis Series B: Real World Applications, 13 (2012), 2476-2490. doi: 10.1016/j.nonrwa.2012.02.013. [30] F. Wu, G. Yin and L. Y. Wang, Stability of a Pure Random Delay System with Two-Time-Scale Markovian Switching, Journal of Differential Equations, 253 (2012), 878-905. doi: 10.1016/j.jde.2012.04.017. [31] G. Yin, Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process, Asymptotic Analysis, 65 (2009), 203-222. [32] G. Yin, S. Kan, L. Y. Wang and C. Xu, Identification of systems with regime switching and unmodeled dynamics, IEEE Transaction on Automatical Control, 54 (2009), 34-47. doi: 10.1109/TAC.2008.2009487. [33] G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Singular Perturbation Approach, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0627-9. [34] G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382. doi: 10.1137/110851171. [35] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [36] C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Processes and their Applications, 103 (2003), 277-291. doi: 10.1016/S0304-4149(02)00230-2. [37] C. Yuan and G. Yin, Stability of hybrid stochastic delay systems whose discrete components have a large state space: A two-time- scale approach, Journal of Mathematical Analysis and Applications, 368 (2010), 103-119. doi: 10.1016/j.jmaa.2010.02.053. [38] Q. Zhang, Hybrid filtering for linear systems with non-Gaussian disturbances, IEEE Transaction on Automatical Control, 45 (2000), 50-61. doi: 10.1109/9.827355. [39] X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583. [40] C. Zhu, G. Yin and Q. S. Song, Stability of random- switching systems of differential equations, Quarterly of Applied Mathematics, 67 (2009), 201-220. doi: 10.1090/S0033-569X-09-01092-8.
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