Abstract
This chapter investigates uncertain dynamical systems under viability (or state) constraints which is the purpose of viability theory. It gathers the concepts and mathematical and algorithmic results addressing this issue.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
They are examples of the viability approach to differential games extensively studied (see for instance [15, Chap. 14, p. 451; 75]).
- 2.
With no consensus on the origine of time, however. “What was God doing before He created the Heavens and the Earth?” asked Augustine of Hippo in his confessions. Is His eternity only forward in time and not backward? Introducing the concepts of temporal windows and exit time function bypasses the question of origin of time, and save the physicists the burden of studying what happened before the “Big Bang”.
- 3.
They are derived from the time-independent version of these concepts [28, Definitions 2.11.2, p. 89]: by introducing the characteristic system
$$\begin{aligned} \left\{ \begin{array}{ll} (i) &{} \delta '(t) = +1\\ (ii) &{} x'(t) = f(\delta (t), x(t), v(t)) \quad \text{ where }\;v(t) \in (\delta (t), x(t)) \end{array} \right. \end{aligned}$$(4.6)the graph of \(\text{ TubAbs }_{f_{[\widetilde{V}]}}(\mathbb {K}, \mathbb {C})[\cdot , D]\) is the (time-independent) absorption basin of the graph of the tube:
$$\begin{aligned} \text{ Graph }(\text{ TubAbs }_{f_{[\widetilde{V}]}}(\mathbb {K}, \mathbb {C})[\cdot , D]) \; := \; \text{ Abs }_{(4.6)}(\text{ Graph }(\mathbb {K}), \{D\}\times C(D)) \end{aligned}$$(4.7)For \(D:=+\infty \), we obtain
$$\begin{aligned} \text{ Graph }(\text{ TubInv }_{f_{[\widetilde{V}]}}(\mathbb {K})) := \text{ Inv }_{(4.6)}(\text{ Graph }(\mathbb {K})) \end{aligned}$$(4.8)Indeed, to say that an element \((d, x)\) belongs to \(\text{ Abs }_{(4.6)}(\text{ Graph }(K); \{D\} \times C(D))\) means that for all evolutions \(t \mapsto (d+t, \overrightarrow{x}(t)) \) where \(\overrightarrow{x}(\cdot ) \) starts at \(\overrightarrow{x}(0)=x\) governed by
$$\begin{aligned} x'(t) = f(d+t, \overrightarrow{x}(t), \overrightarrow{v}(t)) \quad {\text {where}}\; \overrightarrow{v}(t) \in V(d+t, \overrightarrow{v}(t)) \end{aligned}$$there exists \(t^{\star } \ge 0\) such that
$$\begin{aligned} \overrightarrow{x}(t^{\star }) \in \{D\} \times C(D) \end{aligned}$$and
$$\begin{aligned} \forall t \in [0, t^{\star }], \quad \overrightarrow{x}(t) \in K(d+t) \end{aligned}$$This means that \(t^{\star }=D-d\). Setting \(x(t):= \overrightarrow{x}(t-d)\) and \(v(t):= \overrightarrow{v}(t-d)\), we infer that \(x(d)=x\), that for all \(t \in [0, D]\), \(x'(t)=f(t, x(t), v(t))\) where \(v(t)\in V(t, x(t))\) and \(x(D)=x\). In other words, that \(x \in \text{ TubAbs }_{(f, V)}(K, C)[d, D]\) and thus, that \((d, x)\) belongs to its graph.
The case when \(D=+\infty \) is obtained when we take for tubular target the empty set, so that we introduce the invariance kernel of the tubular environment and observe that in the above proof, \(t^{\star }=+\infty \). \(\blacksquare \)
- 4.
see for instance [11, 18, 134].
- 5.
Graphical derivatives of set-valued maps had been introduced in [12] (1981) as an adaptation to set-valued maps of the Fermat geometrical definition of derivatives: the graph of derivative is the tangent cone to the graph. By lack of space, we gave directly the Leibniz analytical version (see [44]: we set here \(\overrightarrow{D}K(t, x):=DK(t, x)(+1)\) and \(\overleftarrow{D}K(t, x):=-DK(t, x)(-1)\)). For governing the evolution of tubes in the same way as differential equations govern the evolutions, the pointwise version of “velocities” of tubes had been introduced in 1992 under the name of transitions and are used to define mutational equations governing the evolution of tubes (see [20], by J.-P. Aubin and [121] by Thomas Lorenz).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Aubin, JP., Chen, L., Dordan, O. (2014). Why Viability Theory? A Survival Kit. In: Tychastic Measure of Viability Risk. Springer, Cham. https://doi.org/10.1007/978-3-319-08129-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-08129-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08128-1
Online ISBN: 978-3-319-08129-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)