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Random Dynamical Systems with Inputs

Part of the Lecture Notes in Mathematics book series (LNMBIOS,volume 2102)

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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Fig. 2.1
Fig. 2.2

Notes

  1. 1.

    Arnold [3, p. 635] and Chueshov [5, p. 10, Definition 1.1.1] refer to such an object primarily as a metric dynamical system. We find measure preserving, which Arnold also uses as a synonym, less confusing and more informative.

  2. 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. 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. 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. 5.

    That is, \(\theta _{t}\tilde{\varOmega } =\tilde{\varOmega }\) for all \(t \in \mathcal{T}\), and \(\mathbb{P}(\tilde{\varOmega }) = 1\).

  6. 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. 7.

    Our convention is that \(\inf \varnothing:= +\infty \).

  8. 8.

    For any \(x \in \mathbb{R}\), we write \(\lceil x\rceil \) to denote the smallest integer larger than or equal to x.

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Acknowledgements

Work supported in part by grants NIH 1R01GM086881 and 1R01GM100473, and AFOSR FA9550-11-1-0247.

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Correspondence to Eduardo D. Sontag .

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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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