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Keywords:
jump parameter system; Markov process; asymptotic stability
Summary:
Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by ${\rm d} X(t) = A(\xi (t))X(t) {\rm d} t + H(\xi (t))X(t) {\rm d} w(t)$, where $\xi (t)$ is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.
References:
[1] Bitmead, R. R., Anderson, B. D. O.: Lyapunov techniques for the exponential stability of linear difference equations with random coefficients. IEEE Trans. Autom. Control 25 (1980), 782-787. DOI 10.1109/TAC.1980.1102427 | MR 0583456 | Zbl 0451.93065
[2] Blair, W. P. Jr., Sworder, D. D.: Continuous-time regulation of a class of econometric models. IEEE Trans. Systems Man Cyber. 5 (1975), 341-346. MR 0469423 | Zbl 0302.90013
[3] Blankenship, G.: Stability of linear differential equations with random coefficients. IEEE Trans. Autom. Control 22 (1977), 834-838. DOI 10.1109/TAC.1977.1101612 | MR 0470386 | Zbl 0362.93033
[4] Blom, H. A. P., Bar-Shalom, Y.: The interacting multiple model algorithm for systems with Markovian switching coefficients. IEEE Trans. Autom. Control 33 (1988), 780-783. DOI 10.1109/9.1299 | Zbl 0649.93065
[5] Costa, O. L. V., Fragoso, M. D.: Stability results for discrete-time linear systems with Markovian jumping parameters. J. Math. Anal. Appl. 179 (1993), 154-178. DOI 10.1006/jmaa.1993.1341 | MR 1244955 | Zbl 0790.93108
[6] Souza, C. E. de, Fragoso, M. D.: $H^\infty$ control for linear systems with Markovian jumping parameters. Control Theory Adv. Tech. 9 (1993), 457-466. MR 1236777
[7] Fang, Y.: A new general sufficient condition for almost sure stability of jump linear systems. IEEE Trans. Autom. Control 42 (1997), 378-382. DOI 10.1109/9.557580 | MR 1435826 | Zbl 0870.93048
[8] Feng, X., Loparo, K. A., Ji, Y., Chizeck, H. J.: Stochastic stability properties of jump linear systems. IEEE Trans. Autom. Control 37 (1992), 1884-1892. DOI 10.1109/9.109637 | MR 1139614 | Zbl 0747.93079
[9] Fragoso, M. D., Costa, O. L. V.: A unified approach for mean square stability of continuous-time linear systems with Markovian jumping parameters and additive disturbances. SIAM J. Control Optim. 44 (2005), 1165-1191. DOI 10.1137/S0363012903434753 | MR 2177308
[10] Ji, Y., Chizeck, H. J.: Controllability, stabilizability, and continuous-time Markovian jumping linear quadratic control. IEEE Trans. Autom. Control 35 (1990), 777-788. DOI 10.1109/9.57016 | MR 1058362
[11] Ji, Y., Chizeck, H. J.: Jump linear quadratic Gaussian control: steady-state solution and testable conditions. Control Theory Adv. Tech. 6 (1990), 289-319. MR 1080011
[12] Leizarowitz, A.: Estimates and exact expressions for Lyapunov exponents of stochastic linear differential equations. Stochastics 24 (1988), 335-356. DOI 10.1080/17442508808833523 | MR 0972970 | Zbl 0669.34060
[13] Loparo, K. A.: Stochastic stability of coupled linear systems: A survey of method and results. Stochastic Anal. Appl. 2 (1984), 193-228. DOI 10.1080/07362998408809033 | MR 0746436
[14] Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006). MR 2256095 | Zbl 1126.60002
[15] Mariton, M.: Almost sure and moments stability of jump linear systems. Systems-Control Letts. 11 (1988), 393-397. DOI 10.1016/0167-6911(88)90098-9 | MR 0971460 | Zbl 0672.93073
[16] Morozan, T.: Optimal stationary control for dynamic systems with Markov perturbations. Stochastic Anal. Appl. 1 (1983), 219-225. DOI 10.1080/07362998308809016 | MR 0709081 | Zbl 0525.93070
[17] Shmerling, E., Hochberg, K. J.: Stability of stochastic jump-parameter semi-Markov linear systems of differential equations. Stochastics 80 (2008), 513-518. MR 2460246 | Zbl 1153.60389
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