Random dynamical systems with inputs. (English) Zbl 1311.37038
Kloeden, Peter E. (ed.) et al., Nonautonomous dynamical systems in the life sciences. Cham: Springer (ISBN 978-3-319-03079-1/pbk; 978-3-319-03080-7/ebook). Lecture Notes in Mathematics 2102. Mathematical Biosciences Subseries, 41-87 (2013).
The basic aim of Chapter 2 in this collective monograph is to propose a new random dynamic system (RDS) formalism for random control systems, i.e., systems with inputs and outputs RDSIO. After a short review of classical RDS theory, the authors introduce the notion of RDSIO extending the notion of RDS to systems in which there is an external input as forcing function, which is itself a stochastic process. Then they turn to the question of “converging input to converging state” (CICS) properties, introduce a class of monotone RDSI, and prove for this class a CICS theorem playing a key role in the stability analysis of feedback interconnections for monotone systems. In a conclusion, a few possible applications of a CICS theorem to life sciences are discussed.
For the entire collection see [Zbl 1282.37004].
For the entire collection see [Zbl 1282.37004].
Reviewer: Irina V. Konopleva (Ul’yanovsk)
MSC:
| 37H05 | General theory of random and stochastic dynamical systems |
| 37H10 | Generation, random and stochastic difference and differential equations |
| 93E15 | Stochastic stability in control theory |
| 93E20 | Optimal stochastic control |
References:
| [1] | Angeli, D.; Sontag, E. D., Monotone control systems, IEEE Trans. Automat. Contr., 48, 10, 1684-1698 (2003) · Zbl 1364.93326 · doi:10.1109/TAC.2003.817920 |
| [2] | Arcak, M.; Sontag, E. D., Diagonal stability for a class of cyclic systems and applications, Automatica, 42, 1531-1537 (2006) · Zbl 1132.39002 · doi:10.1016/j.automatica.2006.04.009 |
| [3] | Arnold, L., Random Dynamical Systems (2010), Berlin: Springer, Berlin |
| [4] | Cao, F.; Jiang, J., On the global attractivity of monotone random dynamical systems, Proc. Am. Math. Soc., 138, 3, 891-898 (2010) · Zbl 1186.37063 · doi:10.1090/S0002-9939-09-09912-2 |
| [5] | I. Chueshov, Monotone Random Systems - Theory and Applications. Lecture Notes in Mathematics, vol. 1997 (Springer, Berlin, 2002) · Zbl 1023.37030 |
| [6] | Crauel, H.; Kloeden, P. E.; Yang, M., Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11, 301-314 (2011) · Zbl 1235.60076 · doi:10.1142/S0219493711003292 |
| [7] | Deng, H.; Krstic, M., Stochastic nonlinear stabilization, Part I: a backstepping design. Syst. Control Lett., 32, 143-150 (1997) · Zbl 0902.93049 |
| [8] | Deng, H.; Krstic, M.; Williams, R., Stabilization of stochastic nonlinear systems driven by noise of unknown covariance, IEEE Trans. Automat. Contr., 46, 1237-1253 (2001) · Zbl 1008.93068 · doi:10.1109/9.940927 |
| [9] | Del Vecchio, D.; Ninfa, A. J.; Sontag, E. D., Modular cell biology: retroactivity and insulation, Nat. Mol. Syst. Biol., 4, 161 (2008) |
| [10] | Enciso, G. A.; Sontag, E. D., Global attractivity, I/O monotone small-gain theorems, and biological delay systems, Discrete Contin. Dyn. Syst., 14, 3, 549-578 (2006) · Zbl 1111.93071 |
| [11] | Florchinger, P., Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method, SIAM J. Control Optim., 35, 500-511 (1997) · Zbl 0874.93092 · doi:10.1137/S0363012995279961 |
| [12] | Folland, G. B., Real Analysis: Modern Techniques and Their Applications (1999), New York: Wiley, New York · Zbl 0924.28001 |
| [13] | T. Gedeon, Cyclic Feedback Systems. Memoirs of the AMS, vol. 134, no. 637 (AMS, Providence, 1998) · Zbl 0991.34051 |
| [14] | Goldbeter, A., Biochemical Oscillations and Cellular Rhythms (1996), Cambridge: Cambridge University Press, Cambridge · Zbl 0837.92009 · doi:10.1017/CBO9780511608193 |
| [15] | Hartwell, L. H.; Hopfield, J. J.; Leibler, S.; Murray, A. W., From molecular to modular cell biology, Nature, 402, 6761, 47-52 (1999) · doi:10.1038/35011540 |
| [16] | S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory (Kluwer, Dordrecht, 1997) · Zbl 0887.47001 |
| [17] | Isidori, A., Nonlinear Control Systems II (1999), London: Springer, London · Zbl 0931.93005 · doi:10.1007/978-1-4471-0549-7 |
| [18] | Jentzen, A.; Kloeden, P. E., Taylor Approximations of Stochastic Partial Differential Equations CBMS Lecture Series (2011), Philadelphia: SIAM, Philadelphia · Zbl 1240.35001 |
| [19] | Khalil, H., Nonlinear Systems (2002), Englewood Cliffs: Prentice Hall, Englewood Cliffs · Zbl 1003.34002 |
| [20] | Krstić, M.; Kanellakopoulos, I.; Kokotović, P. V., Nonlinear and Adaptive Control Design (1995), New York: Wiley, New York · Zbl 0763.93043 |
| [21] | Lang, S., Real Analysis (1983), Reading: Addison-Wesley, Reading · Zbl 0502.46003 |
| [22] | Lauffenburger, D. A., Cell signaling pathways as control modules: complexity for simplicity?, Proc. Natl. Acad. Sci. USA, 97, 10, 5031-5033 (2000) · doi:10.1073/pnas.97.10.5031 |
| [23] | Lindgren, G., Stationary Stochastic Processes—Theory and Applications (2012), London: Chapman and Hall, London |
| [24] | Mallet-Paret, J.; Smith, H. L., The Poincaré-Bendixson theorem for monotone cyclic feedback systems, J. Dyn. Differ. Equ., 2, 367-421 (1990) · Zbl 0712.34060 · doi:10.1007/BF01054041 |
| [25] | Murray, J. D., Mathematical Biology, I, II: An Introduction (2002), New York: Springer, New York · Zbl 1006.92001 |
| [26] | Pan, Z.; Bassar, T., Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion, SIAM J. Control Optim., 37, 957-995 (1999) · Zbl 0924.93046 · doi:10.1137/S0363012996307059 |
| [27] | Smith, H. L., Oscillations and multiple steady states in a cyclic gene model with repression, J. Math. Biol., 25, 169-190 (1987) · Zbl 0619.92004 · doi:10.1007/BF00276388 |
| [28] | H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41 (AMS, Providence, 1995) · Zbl 0821.34003 |
| [29] | Sontag, E. D., Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Contr., 34, 4, 435-443 (1989) · Zbl 0682.93045 · doi:10.1109/9.28018 |
| [30] | E.D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2nd edn. Texts in Applied Mathematics, vol. 6 (Springer, New York, 1998) · Zbl 0945.93001 |
| [31] | Tsinias, J., The concept of ‘exponential ISS’ for stochastic systems and applications to feedback stabilization, Syst. Control Lett., 36, 221-229 (1999) · Zbl 0913.93067 · doi:10.1016/S0167-6911(98)00095-4 |
| [32] | Walters, P., An Introduction to Ergodic Theory (2000), Berlin: Springer, Berlin · Zbl 0958.28011 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
