Appendix A. Background on viability theory
In its original deterministic version [28], viability theory deals with controlled systems such that wðtÞ 0, and for which full information is available: yðtÞ¼ðxðtÞ;πÞ. Eq. (16) can be simplified into yðtþ1Þ¼fðt;yðtÞ;uðtÞÞ ðA:1Þ In this framework, a trajectoryis defined byan initial state y0 and a strategy uð Þ, so the state can be noted yðt;y0;uð ÞÞ. The central question of viability is whether that trajectory leaves a survival set S(t), at any given date within the time frame [0,T]. An answer to this question is brought about by a central object, the viability kernel, which is the setof all initial states for which the system can be controlled so its trajectory does not leave the survival set: ViabðTÞ¼ y0AYj(uð ÞAUðTÞ; 8tA½0;T ;yðt;y0;uð ÞÞASðtÞ ðA:2Þ Thus, an initial state can either be viable or not, which we can translate into reliability terms by stating that in a deterministic context, the probability of failure is either 1 when y0AViabðTÞ, and
0 otherwise. Properties of the viability kernel have provided the foundation of viability algorithms. This is for instance the case for algorithms that use the binary nature of a state under determi- nistic viability (e.g. [30,32]), or the fact that viable trajectories are tangent to the surface of the viability kernel [31]. An interest of these algorithms is that they find both the viable initial states and the associated viable controls.
Appendix B. Computation of pfðt;y0;uð ÞÞ for toT and a fixed uð Þ Inwhat follows we assume a predefined value of uðt;yÞfor eacht and y.
B.1. Forward
This is done through the direct computation of the possible trajectories xðt;y0;uð Þ;wð ÞÞ for all dates 0otrT, as long as they do not leave the survival set. We recursively compute the value function V1ðt;y0;uð Þ;ykÞ: V1ðt;y0;uð Þ;ykÞ¼Pðfyðt;y0;uð Þ;wð ÞÞ¼ykg f8τot;Xðt;πÞASðτÞjyÞ;uð ÞgÞ ðB:1Þ The value function V1 gives the probability of transitioning from y0 at the initial date to yk at date t, while keeping the system in the survival set. In other words, we have pfðt;y0;uð ÞÞ¼1 ∑ yk AY V1ðt;y0;uð Þ;ykÞð B:2Þ The value function V1 is computed through a forward iterative scheme. Initialization reads:
V1ð0;y0;uð Þ;ykÞ¼
PðXð0;πÞASð0Þjy0Þ if y0 ¼yk 0 if y0ayk( ðB:3Þ then the function V1 is recursively updated at each date 1rtrT:
V1ðt;y0;uð Þ;ykÞ¼ ∑ yi AY
V1ðt 1;y0;uð Þ;yiÞ Pðfðt 1;yi;uðt 1;yiÞ;wðt 1ÞÞ¼ykÞ ! PðXðt;πÞASðtÞjykÞ ðB:4Þ The advantage of using the above approach is that it yields the failure probabilities at all dates recursively, in a single run. The inconvenient lies with the large amount of computational memory it requires, since it connects all the points of the successive survival sets with each other.
B.2. Backward
Let us introduce a value function V2 to compute the probability of not failing between the initial date and set a date tA½0;T . V2 is to be computed recursively backwards from t to 0. It is initialized through: V2ðt;yiÞ¼PðXðt;πÞASðtÞjyiÞð B:5Þ and then for τA½0;τ½, the backward transition equation reads: V2ðτ;yiÞ¼ ∑ yk AY Pðfðt;yi;uðτ;yiÞ;wðτÞÞ¼ykÞ V2ðτþ1;ykÞ ! PðXðτ;πÞASðτÞjyiÞð B:6Þ These equations are exact analogous to Eqs. (24) and (25) for the value function V, where there is only one possible control uðτ;yiÞ at each date τ and state yi. Thus, Eq. (26) becomes V2ð0;yiÞ¼ 1 pfðt;y0;uð ÞÞ. This is less expensive than the algorithm for V since there is no need to solve an optimization problem at each date and state to get the feedbacks. However, it is necessary to run this algorithm for each date separately so as to get the probability of failure at multiple dates.
Appendix A. Background on viability theory
In its original deterministic version [28], viability theory deals with controlled systems such that wðtÞ 0, and for which full information is available: yðtÞ¼ðxðtÞ;πÞ. Eq. (16) can be simplified into yðtþ1Þ¼fðt;yðtÞ;uðtÞÞ ðA:1Þ In this framework, a trajectoryis defined byan initial state y0 and a strategy uð Þ, so the state can be noted yðt;y0;uð ÞÞ. The central question of viability is whether that trajectory leaves a survival set S(t), at any given date within the time frame [0,T]. An answer to this question is brought about by a central object, the viability kernel, which is the setof all initial states for which the system can be controlled so its trajectory does not leave the survival set: ViabðTÞ¼ y0AYj(uð ÞAUðTÞ; 8tA½0;T ;yðt;y0;uð ÞÞASðtÞ ðA:2Þ Thus, an initial state can either be viable or not, which we can translate into reliability terms by stating that in a deterministic context, the probability of failure is either 1 when y0AViabðTÞ, and
0 otherwise. Properties of the viability kernel have provided the foundation of viability algorithms. This is for instance the case for algorithms that use the binary nature of a state under determi- nistic viability (e.g. [30,32]), or the fact that viable trajectories are tangent to the surface of the viability kernel [31]. An interest of these algorithms is that they find both the viable initial states and the associated viable controls.
Appendix B. Computation of pfðt;y0;uð ÞÞ for toT and a fixed uð Þ Inwhat follows we assume a predefined value of uðt;yÞfor eacht and y.
B.1. Forward
This is done through the direct computation of the possible trajectories xðt;y0;uð Þ;wð ÞÞ for all dates 0otrT, as long as they do not leave the survival set. We recursively compute the value function V1ðt;y0;uð Þ;ykÞ: V1ðt;y0;uð Þ;ykÞ¼Pðfyðt;y0;uð Þ;wð ÞÞ¼ykg f8τot;Xðt;πÞASðτÞjyÞ;uð ÞgÞ ðB:1Þ The value function V1 gives the probability of transitioning from y0 at the initial date to yk at date t, while keeping the system in the survival set. In other words, we have pfðt;y0;uð ÞÞ¼1 ∑ yk AY V1ðt;y0;uð Þ;ykÞð B:2Þ The value function V1 is computed through a forward iterative scheme. Initialization reads:
V1ð0;y0;uð Þ;ykÞ¼
PðXð0;πÞASð0Þjy0Þ if y0 ¼yk 0 if y0ayk( ðB:3Þ then the function V1 is recursively updated at each date 1rtrT:
V1ðt;y0;uð Þ;ykÞ¼ ∑ yi AY
V1ðt 1;y0;uð Þ;yiÞ Pðfðt 1;yi;uðt 1;yiÞ;wðt 1ÞÞ¼ykÞ ! PðXðt;πÞASðtÞjykÞ ðB:4Þ The advantage of using the above approach is that it yields the failure probabilities at all dates recursively, in a single run. The inconvenient lies with the large amount of computational memory it requires, since it connects all the points of the successive survival sets with each other.
B.2. Backward
Let us introduce a value function V2 to compute the probability of not failing between the initial date and set a date tA½0;T . V2 is to be computed recursively backwards from t to 0. It is initialized through: V2ðt;yiÞ¼PðXðt;πÞASðtÞjyiÞð B:5Þ and then for τA½0;τ½, the backward transition equation reads: V2ðτ;yiÞ¼ ∑ yk AY Pðfðt;yi;uðτ;yiÞ;wðτÞÞ¼ykÞ V2ðτþ1;ykÞ ! PðXðτ;πÞASðτÞjyiÞð B:6Þ These equations are exact analogous to Eqs. (24) and (25) for the value function V, where there is only one possible control uðτ;yiÞ at each date τ and state yi. Thus, Eq. (26) becomes V2ð0;yiÞ¼ 1 pfðt;y0;uð ÞÞ. This is less expensive than the algorithm for V since there is no need to solve an optimization problem at each date and state to get the feedbacks. However, it is necessary to run this algorithm for each date separately so as to get the probability of failure at multiple dates.
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