Abstract
The viability kernel in Viability Theory depends on control variables and usually also on uncontrolled ones. Control variables try to increase viability, and uncontrolled ones instead destroy it. Tyches are uncertainties without statistical regularity that diminish viability. We progress in the study of both effects. We use a necessary condition of the system viability and apply it to the linear case by introducing the Minkowski difference between sets. We also find such a difference interprets the problem adequately.
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Notes
- 1.
Functions \(x:[0,T]\rightarrow \mathcal {H}\) such that \(\Vert x\Vert \) and \(\Vert \dot{x}\Vert \) are integrable.
- 2.
We will see later why this quantity was chosen.
- 3.
Let \(\mathcal {X}\) be a set, \(g:\mathcal {X}\rightarrow [-\infty ,+\infty ]\) be a function, and \(\xi \in \mathbb {R}\). The lower level set of g at height \(\xi \) is the set \(\text {lev}_{\le \xi }\, g\doteq \{x\in \mathcal {X}:g(x)\le \xi \}\) .
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Laengle, S., Laengle-Aliaga, T. (2023). Decreasing Viability of Tychastic Controlled Systems. In: Grothe, O., Nickel, S., Rebennack, S., Stein, O. (eds) Operations Research Proceedings 2022. OR 2022. Lecture Notes in Operations Research. Springer, Cham. https://doi.org/10.1007/978-3-031-24907-5_44
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