• Previous Article
    Robust optimal investment and reinsurance of an insurer under Jump-diffusion models
  • MCRF Home
  • This Issue
  • Next Article
    Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence
March  2019, 9(1): 39-58. doi: 10.3934/mcrf.2019002

Decay rates for stabilization of linear continuous-time systems with random switching

1. 

Institut für Mathematik, Universität Augsburg, 86159 Augsburg, Germany

2. 

Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France

* Corresponding author: Guilherme Mazanti

Received  May 2017 Revised  March 2018 Published  August 2018

Fund Project: The first author is supported by DFG grant Co 124/19-1. The second author is partially supported by the iCODE Institute, research project of the IDEX Paris-Saclay, and by the Hadamard Mathematics LabEx (LMH) through the grant number ANR-11-LABX-0056-LMH in the "Programme des Investissements d'Avenir". This work was prepared while the second author was with CMAP & Inria, team GECO, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France.

For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.

Citation: Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control & Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, 25 (2012), 2937-2952.  doi: 10.1088/0951-7715/25/10/2937.  Google Scholar

[3]

M. BenaïmS. Le BorgneF. Malrieu and P.-A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 1040-1075.  doi: 10.1214/14-AIHP619.  Google Scholar

[4]

P. BolzernP. Colaneri and G. De Nicolao, On almost sure stability of continuous-time Markov jump linear systems, Automatica J. IFAC, 42 (2006), 983-988.  doi: 10.1016/j.automatica.2006.02.007.  Google Scholar

[5]

A. ChailletY. ChitourA. Loría and M. Sigalotti, Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases, Math. Control Signals Systems, 20 (2008), 135-156.  doi: 10.1007/s00498-008-0024-1.  Google Scholar

[6]

D. Chatterjee and D. Liberzon, On stability of randomly switched nonlinear systems, IEEE Trans. Automat. Control, 52 (2007), 2390-2394.  doi: 10.1109/TAC.2007.904253.  Google Scholar

[7]

D. ChengL. GuoY. Lin and Y. Wang, A note on overshoot estimation in pole placements, J. Control Theory Appl., 2 (2004), 161-164.  doi: 10.1007/s11768-004-0062-2.  Google Scholar

[8]

D. ChengL. GuoY. Lin and Y. Wang, Erratum to: "A note on overshoot estimation in pole placements" [J. Control Theory App., 2 (2004), 161-164], J. Control Theory Appl., 3 (2005), 258.  doi: 10.1007/s11768-004-0062-2.  Google Scholar

[9]

Y. ChitourF. Colonius and M. Sigalotti, Growth rates for persistently excited linear systems, Math. Control Signals Systems, 26 (2014), 589-616.  doi: 10.1007/s00498-014-0131-0.  Google Scholar

[10]

Y. Chitour, G. Mazanti and M. Sigalotti, Stabilization of persistently excited linear systems, in Hybrid Systems with Constraints (eds. J. Daafouz, S. Tarbouriech and M. Sigalotti), Wiley-ISTE, London, UK, 2013, chapter 4. doi: 10.1002/9781118639856.ch4.  Google Scholar

[11]

Y. Chitour and M. Sigalotti, On the stabilization of persistently excited linear systems, SIAM J. Control Optim., 48 (2010), 4032-4055.  doi: 10.1137/080737812.  Google Scholar

[12]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.  Google Scholar

[13]

O. L. V. Costa, M. D. Fragoso and M. G. Todorov, Continuous-Time Markov Jump Linear Systems, Probability and its Applications (New York), Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-34100-7.  Google Scholar

[14]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388.   Google Scholar

[15]

M. H. A. Davis, Markov Models and Optimization, vol. 49 of Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1993. doi: 10.1007/978-1-4899-4483-2.  Google Scholar

[16]

A. DiwadkarS. Dasgupta and U. Vaidya, Control of systems in Lure form over erasure channels, Internat. J. Robust Nonlinear Control, 25 (2015), 2787-2802.  doi: 10.1002/rnc.3231.  Google Scholar

[17]

A. Diwadkar and U. Vaidya, Stabilization of linear time varying systems over uncertain channels, Internat. J. Robust Nonlinear Control, 24 (2014), 1205-1220.  doi: 10.1002/rnc.2935.  Google Scholar

[18]

Y. Fang and K. A. Loparo, Stabilization of continuous-time jump linear systems, IEEE Trans. Automat. Control, 47 (2002), 1590-1603.  doi: 10.1109/TAC.2002.803528.  Google Scholar

[19]

X. FengK. A. LoparoY. Ji and H. J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control, 37 (1992), 38-53.  doi: 10.1109/9.109637.  Google Scholar

[20]

J. J. Green, Uniform Convergence to the Spectral Radius and Some Related Properties in Banach Algebras, PhD thesis, University of Sheffield, 1996. Google Scholar

[21]

X. GuyonS. Iovleff and J.-F. Yao, Linear diffusion with stationary switching regime, ESAIM Probab. Stat., 8 (2004), 25-35.  doi: 10.1051/ps:2003017.  Google Scholar

[22]

M. Hairer, Ergodic Properties of Markov Processes, Lecture notes from the University of Warwick, 2006. Google Scholar

[23]

P. R. Halmos, Measure Theory, vol. 18 of Graduate Texts in Mathematics, Springer-Verlag, 1974. doi: 10.1007/978-1-4684-9440-2.  Google Scholar

[24]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013. doi: 10.1017/cbo9781139020411.  Google Scholar

[25]

C. LiM. Z. Q. ChenJ. Lam and X. Mao, On exponential almost sure stability of random jump systems, IEEE Trans. Automat. Control, 57 (2012), 3064-3077.  doi: 10.1109/TAC.2012.2200369.  Google Scholar

[26]

D. Liberzon, Switching in Systems and Control, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0017-8.  Google Scholar

[27]

H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), 308-322.  doi: 10.1109/TAC.2008.2012009.  Google Scholar

[28]

M. Margaliot, Stability analysis of switched systems using variational principles: an introduction, Automatica, 42 (2006), 2059-2077.  doi: 10.1016/j.automatica.2006.06.020.  Google Scholar

[29]

S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.  Google Scholar

[30]

J. R. Norris, Markov Chains, vol. 2 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998, Reprint of 1997 original. doi: 10.1017/cbo9780511810633.  Google Scholar

[31]

W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, New York, 1987.  Google Scholar

[32]

R. ShortenF. WirthO. MasonK. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.  doi: 10.1137/05063516X.  Google Scholar

[33]

S. Srikant and M. R. Akella, Arbitrarily fast exponentially stabilizing controller for multi-input, persistently exciting singular control gain systems, Automatica J. IFAC, 54 (2015), 279-283.  doi: 10.1016/j.automatica.2015.02.008.  Google Scholar

[34]

Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Communications and Control Engineering, Springer-Verlag, London, 2005. doi: 10.1007/1-84628-131-8.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, 25 (2012), 2937-2952.  doi: 10.1088/0951-7715/25/10/2937.  Google Scholar

[3]

M. BenaïmS. Le BorgneF. Malrieu and P.-A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 1040-1075.  doi: 10.1214/14-AIHP619.  Google Scholar

[4]

P. BolzernP. Colaneri and G. De Nicolao, On almost sure stability of continuous-time Markov jump linear systems, Automatica J. IFAC, 42 (2006), 983-988.  doi: 10.1016/j.automatica.2006.02.007.  Google Scholar

[5]

A. ChailletY. ChitourA. Loría and M. Sigalotti, Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases, Math. Control Signals Systems, 20 (2008), 135-156.  doi: 10.1007/s00498-008-0024-1.  Google Scholar

[6]

D. Chatterjee and D. Liberzon, On stability of randomly switched nonlinear systems, IEEE Trans. Automat. Control, 52 (2007), 2390-2394.  doi: 10.1109/TAC.2007.904253.  Google Scholar

[7]

D. ChengL. GuoY. Lin and Y. Wang, A note on overshoot estimation in pole placements, J. Control Theory Appl., 2 (2004), 161-164.  doi: 10.1007/s11768-004-0062-2.  Google Scholar

[8]

D. ChengL. GuoY. Lin and Y. Wang, Erratum to: "A note on overshoot estimation in pole placements" [J. Control Theory App., 2 (2004), 161-164], J. Control Theory Appl., 3 (2005), 258.  doi: 10.1007/s11768-004-0062-2.  Google Scholar

[9]

Y. ChitourF. Colonius and M. Sigalotti, Growth rates for persistently excited linear systems, Math. Control Signals Systems, 26 (2014), 589-616.  doi: 10.1007/s00498-014-0131-0.  Google Scholar

[10]

Y. Chitour, G. Mazanti and M. Sigalotti, Stabilization of persistently excited linear systems, in Hybrid Systems with Constraints (eds. J. Daafouz, S. Tarbouriech and M. Sigalotti), Wiley-ISTE, London, UK, 2013, chapter 4. doi: 10.1002/9781118639856.ch4.  Google Scholar

[11]

Y. Chitour and M. Sigalotti, On the stabilization of persistently excited linear systems, SIAM J. Control Optim., 48 (2010), 4032-4055.  doi: 10.1137/080737812.  Google Scholar

[12]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.  Google Scholar

[13]

O. L. V. Costa, M. D. Fragoso and M. G. Todorov, Continuous-Time Markov Jump Linear Systems, Probability and its Applications (New York), Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-34100-7.  Google Scholar

[14]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388.   Google Scholar

[15]

M. H. A. Davis, Markov Models and Optimization, vol. 49 of Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1993. doi: 10.1007/978-1-4899-4483-2.  Google Scholar

[16]

A. DiwadkarS. Dasgupta and U. Vaidya, Control of systems in Lure form over erasure channels, Internat. J. Robust Nonlinear Control, 25 (2015), 2787-2802.  doi: 10.1002/rnc.3231.  Google Scholar

[17]

A. Diwadkar and U. Vaidya, Stabilization of linear time varying systems over uncertain channels, Internat. J. Robust Nonlinear Control, 24 (2014), 1205-1220.  doi: 10.1002/rnc.2935.  Google Scholar

[18]

Y. Fang and K. A. Loparo, Stabilization of continuous-time jump linear systems, IEEE Trans. Automat. Control, 47 (2002), 1590-1603.  doi: 10.1109/TAC.2002.803528.  Google Scholar

[19]

X. FengK. A. LoparoY. Ji and H. J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control, 37 (1992), 38-53.  doi: 10.1109/9.109637.  Google Scholar

[20]

J. J. Green, Uniform Convergence to the Spectral Radius and Some Related Properties in Banach Algebras, PhD thesis, University of Sheffield, 1996. Google Scholar

[21]

X. GuyonS. Iovleff and J.-F. Yao, Linear diffusion with stationary switching regime, ESAIM Probab. Stat., 8 (2004), 25-35.  doi: 10.1051/ps:2003017.  Google Scholar

[22]

M. Hairer, Ergodic Properties of Markov Processes, Lecture notes from the University of Warwick, 2006. Google Scholar

[23]

P. R. Halmos, Measure Theory, vol. 18 of Graduate Texts in Mathematics, Springer-Verlag, 1974. doi: 10.1007/978-1-4684-9440-2.  Google Scholar

[24]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013. doi: 10.1017/cbo9781139020411.  Google Scholar

[25]

C. LiM. Z. Q. ChenJ. Lam and X. Mao, On exponential almost sure stability of random jump systems, IEEE Trans. Automat. Control, 57 (2012), 3064-3077.  doi: 10.1109/TAC.2012.2200369.  Google Scholar

[26]

D. Liberzon, Switching in Systems and Control, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0017-8.  Google Scholar

[27]

H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), 308-322.  doi: 10.1109/TAC.2008.2012009.  Google Scholar

[28]

M. Margaliot, Stability analysis of switched systems using variational principles: an introduction, Automatica, 42 (2006), 2059-2077.  doi: 10.1016/j.automatica.2006.06.020.  Google Scholar

[29]

S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.  Google Scholar

[30]

J. R. Norris, Markov Chains, vol. 2 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998, Reprint of 1997 original. doi: 10.1017/cbo9780511810633.  Google Scholar

[31]

W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, New York, 1987.  Google Scholar

[32]

R. ShortenF. WirthO. MasonK. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.  doi: 10.1137/05063516X.  Google Scholar

[33]

S. Srikant and M. R. Akella, Arbitrarily fast exponentially stabilizing controller for multi-input, persistently exciting singular control gain systems, Automatica J. IFAC, 54 (2015), 279-283.  doi: 10.1016/j.automatica.2015.02.008.  Google Scholar

[34]

Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Communications and Control Engineering, Springer-Verlag, London, 2005. doi: 10.1007/1-84628-131-8.  Google Scholar

[1]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[2]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[3]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304

[4]

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

[5]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[6]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[7]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329

[8]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[9]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[10]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[11]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[12]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[13]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332

[14]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[15]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[16]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[17]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[18]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[19]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304

[20]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (138)
  • HTML views (818)
  • Cited by (2)

Other articles
by authors

[Back to Top]