Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 63, 2014 - Issue 11
144
Views
7
CrossRef citations to date
0
Altmetric
Articles

Stability, stabilizability and detectability for Markov jump discrete-time linear systems with multiplicative noise in Hilbert spaces

Pages 1689-1712 | Received 31 Jan 2012, Accepted 09 Sep 2012, Published online: 06 Dec 2012

References

  • Arnold , L , Oeljeklaus , E and Pardoux , É . 1986 . Almost Sure and Moment Stability for Linear Itô Equations: Lyapunov exponents, Lecture Notes in Mathematics , Vol. 1186 , 129 – 159 . Berlin : Springer .
  • Costa , EF and do Val , JBR . 2001 . On the detectability and observability of discrete-time Markovian jump linear systems . Syst. Control Lett. , 44 : 135 – 145 .
  • Costa , EF and do Val , JBR . 2005 . On the observability and detectability of continuous-time Markov jump linear systems . SIAM J. Control Optim. , 41 : 1295 – 1314 .
  • E.F., Costa, J.B.R. do Val, M.D. Fragoso, A new approach to detectability of discrete-time infinite Markov jump linear systems, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, December 12–15, 2005, pp. 6662–6667
  • Costa , OLV and Fragoso , MD . 1995 . Discrete-time LQ-optimal control problems for infinite Markov jump parameter systems . IEEE Trans. Auto. Contr. , 40 : 2076 – 2088 .
  • Costa , OLV and Fragoso , MD . 1996 . Full information H∞-control for discrete-time infinite Markov jump parameters systems . J. Math. Anal. Appl. , 202 : 578 – 603 .
  • Damm , T . 2007 . On detectability of stochastic systems . Automatica , 43 : 928 – 933 .
  • Dragan , V and Morozan , T . 2006 . Mean square exponential stability for some stochastic linear discrete time systems . Eur. J. Control , 4 : 373 – 395 .
  • Dragan , V and Morozan , T . 2006 . Exponential stability for discrete time linear equations defined by positive operators . Integral Equ. Oper. Theory , 54 : 465 – 493 .
  • Dragan , V and Morozan , T . 2008 . Discrete-time linear equations defined by positive operators on ordered Hilbert spaces . Rev. Roum. Math. Pures Appl. , 53 : 131 – 166 .
  • Dragan , V and Morozan , T . 2009 . Linear quadratic optimization problems for some discrete-time stochastic linear systems . Math. Rep. , 11 : 307 – 319 .
  • Dragan , V and Morozan , T . 2010 . A class of discrete time generalized Riccati equations . J. Differ. Eqns. Appl. , 16 : 291 – 320 .
  • Dragan , V , Morozan , T and Stoica , A . 2010 . Mathematical Methods in Robust Control of Discrete Time Linear Stochastic Systems , New York : Springer .
  • Fragoso , MD and Baczynski , J . 2001 . Optimal control for continuous-time linear quadratic problems with infinite Markov jump parameters . SIAM J. Control Optim. , 40 : 270 – 297 .
  • Fragoso , MD and Baczynski , J . 2002 . Stochastic versus mean square stability in continuous time linear infinite Markov jump parameter systems . Stochastic Anal. Appl. , 20 : 347 – 356 .
  • Fragoso , MD and Baczynski , J . 2002 . Lyapunov coupled equations for continuous-time infinite Markov jump linear systems . J. Math. Anal. Appl. , 274 : 319 – 335 .
  • Gohberg , I and Goldberg , S . 1981 . Basic Operator Theory , Basel : Birkhauser .
  • Harchol-Balter , M , Ogosami , T , Scheller-Wolf , A and Wierman , A . 2005 . Multi-server queueing systems with multiple priority classes . Queueing Syst. , 51 : 331 – 360 .
  • Has'minskiĭ , RZ . 1980 . Stochastic Stability of Differential Equations (Translated from the Russian by D. Louvish) , Vol. 7 , Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, MD : Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis . xvi+344 pp
  • Ji , Y , Chizeck , HJ , Feng , X and Loparo , KA . 1991 . Stability and control of discrete-time jump linear systems . Control Theory Adv. Techn. , 7 : 247 – 270 .
  • Li , ZY , Wang , Y , Zhou , B and Duan , GR . 2009 . Detectability and observability of discrete-time stochastic systems and their applications . Automatica , 45 : 1340 – 1346 .
  • Morozan , T . 1992 . Discrete time Riccati equations connected with quadratic control for linear systems with independent random perturbations . Rev. Roum. Math. Pures Appl. , 37 : 233 – 246 .
  • Morozan , T . 1995 . Stability and control for linear discrete-time systems with Markov perturbations . Rev. Roum. Math. Pures Appl. , 40 : 471 – 494 .
  • Morozan , T . 1997 . Stability radii of some discrete-time systems with independent random perturbations . Stochastic Anal. Appl. , 15 : 375 – 386 .
  • Ni , Y , Zhang , W and Fang , H . 2010 . On the observability and detectability of linear stochastic systems with Markov jumps and multiplicative noise . J. Syst. Sci. Complex , 23 : 102 – 115 .
  • Schaefer , HH . 1974 . Banach Lattices and Positive Operators , New York : Springer-Verlag .
  • Todorov , MG and Fragoso , MD . 2008 . Output feedback H∞ control of continuous-time infinite Markovian jump linear systems via LMI methods . SIAM J. Control Optim. , 47 : 950 – 970 .
  • Todorov , MG and Fragoso , MD . 2008 . Infinite Markov jump-bounded real lemma . Syst. Control Lett. , 57 : 64 – 70 .
  • Ungureanu , VM . 2004 . Uniform exponential stability for linear discrete time systems with stochastic perturbations in Hilbert spaces . Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. , 8 : 757 – 772 .
  • Ungureanu , VM . 2004 . Representations of mild solutions of time-varying linear stochastic equations and the exponential stability of periodic systems . Electr. J. Qual. Theory Differ. Eqns. , 4 : 1 – 22 .
  • Ungureanu , VM . 2005 . Quadratic control problem for linear discrete -time varying systems with multiplicative noise in Hilbert spaces . Math. Rep. No. , 17 : 73 – 89 .
  • Ungureanu , VM . 2009 . Optimal control for linear discrete time systems with Markov perturbations in Hilbert spaces . IMA J. Math. Control Inform. , 26 : 105 – 127 .
  • Ungureanu , VM and Cheng , SS . 2008 . Mean stability of a stochastic difference equation . Ann. Pol. Math. , 1 : 33 – 52 .
  • Ungureanu , VM and Cheng , SS . 2010 . Mean square error synchronization in networks with ring structure . Taiwan. J. Math. , 14 : 2405 – 2433 .
  • Ungureanu , VM and Dragan , V . Stability of discrete-time positive evolution operators on ordered Banach spaces and applications . J. Differ. Eqns. Appl. , DOI: 10.1080/10236198.2012.704369. Available at http://www.tandfonline.com/doi/abs/10.1080/10236198.2012.704369
  • Ungureanu , VM , Dragan , V and Morozan , T . Global solutions of a class of discrete-time backward nonlinear equations on ordered Banach spaces with applications to Riccati equations of stochastic control . Optimal Control Appl. Methods , DOI: 10.1002/oca.2015. Available at http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-1514/earlyview
  • Zabczyk , J . 1975 . On optimal stochastic control of discrete-time systems in Hilbert spaces . SIAM J. Control , 13 : 1217 – 1234 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.