Abstract
This work introduces a notion of random dynamical systems with inputs, providing several basic definitions and results on equilibria and convergence. It also presents a “converging input to converging state” (“CICS”) result, a concept that plays a key role in the analysis of stability of feedback interconnections, for monotone systems.
Keywords
- Pullback convergence
- Random dynamical systems
- Stochastic dynamics
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Notes
- 1.
- 2.
Property (T3) is normally [32, Definition 1.1] stated as
$$\displaystyle{\mathbb{P}(\theta _{t}^{-1}(B)) = \mathbb{P}(B),\quad \forall B \in \mathcal{F},\ \forall t \in \mathcal{T}.}$$But since it follows from (T2) that θ t is invertible with \(\theta _{t}^{-1} =\theta _{-t}\) for each \(t \in \mathcal{T}\), the two formulations are equivalent in this context.
- 3.
The reason we are introducing the fundamental matrix solution as a function of \((s,t) \in \mathbb{R} \times \mathbb{R}\) rather than a function of just \(t \in \mathbb{R}\) (for each fixed ω ∈ Ω) will become clear in Example 2.3. This notation will make it easier to discuss the rate of growth of the fundamental matrix solution.
- 4.
A “θ-stochastic process” is indeed a stochastic process in the traditional sense. We use the prefix “θ-” to emphasize the underlying probability space, as well as the time semigroup.
- 5.
That is, \(\theta _{t}\tilde{\varOmega } =\tilde{\varOmega }\) for all \(t \in \mathcal{T}\), and \(\mathbb{P}(\tilde{\varOmega }) = 1\).
- 6.
We will use the same notation ρ s for the shift operator \(\mathcal{S}_{\theta }^{V } \rightarrow \mathcal{S}_{\theta }^{V }\) defined by (2.5), irrespective of the underlying metric space V. Since the domain of any θ-stochastic process is always \(\mathcal{T}_{\geq 0}\times \varOmega\), this will not be a source of confusion.
- 7.
Our convention is that \(\inf \varnothing:= +\infty \).
- 8.
For any \(x \in \mathbb{R}\), we write \(\lceil x\rceil \) to denote the smallest integer larger than or equal to x.
References
D. Angeli, E.D. Sontag, Monotone control systems. IEEE Trans. Automat. Contr. 48(10), 1684–1698 (2003)
M. Arcak, E.D. Sontag, Diagonal stability for a class of cyclic systems and applications. Automatica 42, 1531–1537 (2006)
L. Arnold, Random Dynamical Systems (Springer, Berlin, 2010)
F. Cao, J. Jiang, On the global attractivity of monotone random dynamical systems. Proc. Am. Math. Soc. 138(3), 891–898 (2010)
I. Chueshov, Monotone Random Systems – Theory and Applications. Lecture Notes in Mathematics, vol. 1997 (Springer, Berlin, 2002)
H. Crauel, P.E. Kloeden, M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains. Stoch. Dyn. 11, 301–314 (2011)
H. Deng, M. Krstic, Stochastic nonlinear stabilization, Part I: a backstepping design. Syst. Control Lett. 32, 143–150 (1997)
H. Deng, M. Krstic, R. Williams, Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Automat. Contr. 46, 1237–1253 (2001)
D. Del Vecchio, A.J. Ninfa, E.D. Sontag, Modular cell biology: retroactivity and insulation. Nat. Mol. Syst. Biol. 4, 161 (2008)
G.A. Enciso, E.D. Sontag, Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete Contin. Dyn. Syst. 14(3), 549–578 (2006)
P. Florchinger, Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method. SIAM J. Control Optim. 35, 500–511 (1997)
G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edn. (Wiley, New York, 1999)
T. Gedeon, Cyclic Feedback Systems. Memoirs of the AMS, vol. 134, no. 637 (AMS, Providence, 1998)
A. Goldbeter, Biochemical Oscillations and Cellular Rhythms (Cambridge University Press, Cambridge, 1996)
L.H. Hartwell, J.J. Hopfield, S. Leibler, A.W. Murray, From molecular to modular cell biology. Nature 402(6761 Suppl), 47–52 (1999)
S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory (Kluwer, Dordrecht, 1997)
A. Isidori, Nonlinear Control Systems II (Springer, London, 1999)
A. Jentzen, P.E. Kloeden, Taylor Approximations of Stochastic Partial Differential Equations CBMS Lecture Series (SIAM, Philadelphia, 2011)
H. Khalil, Nonlinear Systems (Prentice Hall, Englewood Cliffs, 2002)
M. Krstić, I. Kanellakopoulos, P.V. Kokotović, Nonlinear and Adaptive Control Design (Wiley, New York, 1995)
S. Lang, Real Analysis, 2nd edn. (Addison-Wesley, Reading, 1983)
D.A. Lauffenburger, Cell signaling pathways as control modules: complexity for simplicity? Proc. Natl. Acad. Sci. USA 97(10), 5031–5033 (2000)
G. Lindgren, Stationary Stochastic Processes—Theory and Applications (Chapman and Hall, London, 2012)
J. Mallet-Paret, H.L. Smith, The Poincaré-Bendixson theorem for monotone cyclic feedback systems. J. Dyn. Differ. Equ. 2, 367–421 (1990)
J.D. Murray, Mathematical Biology, I, II: An Introduction (Springer, New York, 2002)
Z. Pan, T. Bassar, Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion. SIAM J. Control Optim. 37, 957–995 (1999)
H.L. Smith, Oscillations and multiple steady states in a cyclic gene model with repression. J. Math. Biol. 25, 169–190 (1987)
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)
E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Contr. 34(4), 435–443 (1989)
E.D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2nd edn. Texts in Applied Mathematics, vol. 6 (Springer, New York, 1998)
J. Tsinias, The concept of ‘exponential ISS’ for stochastic systems and applications to feedback stabilization. Syst. Control Lett. 36, 221–229 (1999)
P. Walters, An Introduction to Ergodic Theory (Springer, Berlin, 2000)
Acknowledgements
Work supported in part by grants NIH 1R01GM086881 and 1R01GM100473, and AFOSR FA9550-11-1-0247.
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de Freitas, M.M., Sontag, E.D. (2013). Random Dynamical Systems with Inputs. In: Kloeden, P., Pötzsche, C. (eds) Nonautonomous Dynamical Systems in the Life Sciences. Lecture Notes in Mathematics(), vol 2102. Springer, Cham. https://doi.org/10.1007/978-3-319-03080-7_2
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